The protean chromatic polynomial Bruce E. Sagan Department of - - PowerPoint PPT Presentation

The protean chromatic polynomial

Bruce E. Sagan Department of Mathematics Michigan State University East Lansing, MI 48824-1027 sagan@math.msu.edu www.math.msu.edu/˜sagan Harvard April 2019

Initial definitions The chromatic polynomial Acyclic orientations and hyperplane arrangements Increasing forests Comments and open questions

Outline

Initial definitions The chromatic polynomial Acyclic orientations and hyperplane arrangements Increasing forests Comments and open questions

“Protean” means changeable. From Proteus, a Greek god of the sea and water.

“Protean” means changeable. From Proteus, a Greek god of the sea and water. Proteus

“Protean” means changeable. From Proteus, a Greek god of the sea and water. “Chromatic” means having to do with color. Proteus

“Protean” means changeable. From Proteus, a Greek god of the sea and water. “Chromatic” means having to do with color. Proteus A colorful annulus

Outline

Initial definitions The chromatic polynomial Acyclic orientations and hyperplane arrangements Increasing forests Comments and open questions

Let G = (V , E) be a finite graph with vertices V and edges E.

Let G = (V , E) be a finite graph with vertices V and edges E. Ex. G = x w u v V = {u, v, w, x} E = {uv, ux, vw, vx}

Let G = (V , E) be a finite graph with vertices V and edges E. Ex. G = x w u v V = {u, v, w, x} E = {uv, ux, vw, vx} A coloring of G is a function c : V → {c1, . . . , ct}.

Let G = (V , E) be a finite graph with vertices V and edges E. Ex. G = x w u v V = {u, v, w, x} E = {uv, ux, vw, vx} A coloring of G is a function c : V → {c1, . . . , ct}. The coloring is proper if uv ∈ E = ⇒ c(u) = c(v).

Let G = (V , E) be a finite graph with vertices V and edges E. Ex. G = x w u v V = {u, v, w, x} E = {uv, ux, vw, vx} A coloring of G is a function c : V → {c1, . . . , ct}. The coloring is proper if uv ∈ E = ⇒ c(u) = c(v). Ex. proper,

Let G = (V , E) be a finite graph with vertices V and edges E. Ex. G = x w u v V = {u, v, w, x} E = {uv, ux, vw, vx} A coloring of G is a function c : V → {c1, . . . , ct}. The coloring is proper if uv ∈ E = ⇒ c(u) = c(v). Ex. proper, not proper,

Let G = (V , E) be a finite graph with vertices V and edges E. Ex. G = x w u v V = {u, v, w, x} E = {uv, ux, vw, vx} A coloring of G is a function c : V → {c1, . . . , ct}. The coloring is proper if uv ∈ E = ⇒ c(u) = c(v). Ex. proper, not proper, The chromatic number of G is χ(G) = smallest t such that there is a proper c : V → {c1, . . . , ct}.

Let G = (V , E) be a finite graph with vertices V and edges E. Ex. G = x w u v V = {u, v, w, x} E = {uv, ux, vw, vx} A coloring of G is a function c : V → {c1, . . . , ct}. The coloring is proper if uv ∈ E = ⇒ c(u) = c(v). Ex. proper, not proper, χ(G) = 3. The chromatic number of G is χ(G) = smallest t such that there is a proper c : V → {c1, . . . , ct}.

Let G = (V , E) be a finite graph with vertices V and edges E. Ex. G = x w u v V = {u, v, w, x} E = {uv, ux, vw, vx} A coloring of G is a function c : V → {c1, . . . , ct}. The coloring is proper if uv ∈ E = ⇒ c(u) = c(v). Ex. proper, not proper, χ(G) = 3. The chromatic number of G is χ(G) = smallest t such that there is a proper c : V → {c1, . . . , ct}.

Theorem (Four Color Theorem, Appel-Haken, 1976)

If G is planar (can be drawn in the plane without edge crossings) then χ(G) ≤ 4.

Kenneth Appel Wolfgang Haken

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2)

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2) = t4 − 4t3 + 5t2 − 2t

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2) = t4 − 4t3 + 5t2 − 2t Note 1. This is a polynomial in t.

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2) = t4 − 4t3 + 5t2 − 2t Note 1. This is a polynomial in t.

• 2. χ(G) is the smallest positive integer with P(G, χ(G)) > 0.

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2) = t4 − 4t3 + 5t2 − 2t Note 1. This is a polynomial in t.

• 2. χ(G) is the smallest positive integer with P(G, χ(G)) > 0.
• 3. P(G, t) need not be a product of factors t − k for integers k.

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2) = t4 − 4t3 + 5t2 − 2t Note 1. This is a polynomial in t.

• 2. χ(G) is the smallest positive integer with P(G, χ(G)) > 0.
• 3. P(G, t) need not be a product of factors t − k for integers k.
• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2) = t4 − 4t3 + 5t2 − 2t Note 1. This is a polynomial in t.

• 2. χ(G) is the smallest positive integer with P(G, χ(G)) > 0.
• 3. P(G, t) need not be a product of factors t − k for integers k.
• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2) = t4 − 4t3 + 5t2 − 2t Note 1. This is a polynomial in t.

• 2. χ(G) is the smallest positive integer with P(G, χ(G)) > 0.
• 3. P(G, t) need not be a product of factors t − k for integers k.
• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2) = t4 − 4t3 + 5t2 − 2t Note 1. This is a polynomial in t.

• 2. χ(G) is the smallest positive integer with P(G, χ(G)) > 0.
• 3. P(G, t) need not be a product of factors t − k for integers k.
• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1

For a positive integer t, the chromatic polynomial of G is P(G) = P(G, t) = # of proper colorings c : V → {c1, . . . , ct}.

• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 t − 2 P(G, t) = t(t − 1)(t − 1)(t − 2) = t4 − 4t3 + 5t2 − 2t Note 1. This is a polynomial in t.

• 2. χ(G) is the smallest positive integer with P(G, χ(G)) > 0.
• 3. P(G, t) need not be a product of factors t − k for integers k.
• Ex. Coloring vertices in the order u, v, w, x gives choices

x w u v t t − 1 t − 1 ?

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted.

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve.

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge.

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e x w u v

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e x w u v G \ e = x w u v

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e x w u v G \ e = x w u v G/e = w u ve

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e x w u v G \ e = x w u v G/e = w u ve

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e x w u v G \ e = x w u v G/e = w u ve

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx.

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e x w u v G \ e = x w u v G/e = w u ve

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx. It suffices to show P(G \ e) = P(G) + P(G/e).

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e x w u v G \ e = x w u v G/e = w u ve

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx. It suffices to show P(G \ e) = P(G) + P(G/e). P(G \ e) = (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) + (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x))

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e x w u v G \ e = x w u v G/e = w u ve

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx. It suffices to show P(G \ e) = P(G) + P(G/e). P(G \ e) = (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) + (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) = P(G)

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e G \ e = G/e =

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx. It suffices to show P(G \ e) = P(G) + P(G/e). P(G \ e) = (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) + (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) = P(G)

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e G \ e = G/e =

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx. It suffices to show P(G \ e) = P(G) + P(G/e). P(G \ e) = (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) + (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) = P(G)

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e G \ e = G/e =

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx. It suffices to show P(G \ e) = P(G) + P(G/e). P(G \ e) = (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) + (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) = P(G) + P(G/e)

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e G \ e = G/e =

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx. It suffices to show P(G \ e) = P(G) + P(G/e). P(G \ e) = (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) + (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) = P(G) + P(G/e)

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e G \ e = G/e =

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx. It suffices to show P(G \ e) = P(G) + P(G/e). P(G \ e) = (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) + (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) = P(G) + P(G/e)

If G = (V , E) is a graph and e ∈ E then let G \ e = G with e deleted. G/e = G with e contracted to a vertex ve. Any multiple edge in G/e is replaced by a single edge. Ex. G = e G \ e = G/e =

Lemma (Deletion-Contraction, DC)

If G = (V , E) is any graph and e ∈ E then P(G, t) = P(G \ e; t) − P(G/e; t).

Proof.

Let e = vx. It suffices to show P(G \ e) = P(G) + P(G/e). P(G \ e) = (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) + (# of proper c : G \ e → {c1, . . . , ct} with c(v) = c(x)) = P(G) + P(G/e) as desired.

P(G, t) = P(G \ e; t) − P(G/e; t).

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m.

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m.

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G)

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn.

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn. If m > 0, then pick e ∈ E.

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn. If m > 0, then pick e ∈ E. Both G \ e and G/e have fewer edges than G.

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn. If m > 0, then pick e ∈ E. Both G \ e and G/e have fewer edges than G. So by DC and induction P(G) = P(G\e)−P(G/e)

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn. If m > 0, then pick e ∈ E. Both G \ e and G/e have fewer edges than G. So by DC and induction P(G) = P(G\e)−P(G/e) = polynomial−polynomial

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn. If m > 0, then pick e ∈ E. Both G \ e and G/e have fewer edges than G. So by DC and induction P(G) = P(G\e)−P(G/e) = polynomial−polynomial = polynomial as desired.

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn. If m > 0, then pick e ∈ E. Both G \ e and G/e have fewer edges than G. So by DC and induction P(G) = P(G\e)−P(G/e) = polynomial−polynomial = polynomial as desired. Ex. P

• e

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn. If m > 0, then pick e ∈ E. Both G \ e and G/e have fewer edges than G. So by DC and induction P(G) = P(G\e)−P(G/e) = polynomial−polynomial = polynomial as desired. Ex. P

• e
• =

P

• −

P

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn. If m > 0, then pick e ∈ E. Both G \ e and G/e have fewer edges than G. So by DC and induction P(G) = P(G\e)−P(G/e) = polynomial−polynomial = polynomial as desired. Ex. P

• e
• =

P

• −

P

• = t(t − 1)3 − t(t − 1)(t − 2)

P(G, t) = P(G \ e; t) − P(G/e; t).

Theorem (Birkhoff, 1912)

For any graph G = (V , E), P(G, t) is a polynomial in t.

Proof.

Let |V | = n, |E| = m. Induct on m. If m = 0 then P(G) = tn. If m > 0, then pick e ∈ E. Both G \ e and G/e have fewer edges than G. So by DC and induction P(G) = P(G\e)−P(G/e) = polynomial−polynomial = polynomial as desired. Ex. P

• e
• =

P

• −

P

• = t(t − 1)3 − t(t − 1)(t − 2)

= t(t − 1)(t2 − 3t + 3).

George David Birkhoff

Outline

Initial definitions The chromatic polynomial Acyclic orientations and hyperplane arrangements Increasing forests Comments and open questions

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or vu.

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|.

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. Ex. G =

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. Ex. G = has orientation O =

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Ex. G = has orientation O =

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O =

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O = which is acyclic.

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O = which is acyclic. # of acyclic orientations of G

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O = which is acyclic. # of acyclic orientations of G = (# for the triangle)(# for the remaining edge)

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O = which is acyclic. # of acyclic orientations of G = (# for the triangle)(# for the remaining edge) = (23 − 2)(2)

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O = which is acyclic. # of acyclic orientations of G = (# for the triangle)(# for the remaining edge) = (23 − 2)(2) = 12.

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O = which is acyclic. # of acyclic orientations of G = (# for the triangle)(# for the remaining edge) = (23 − 2)(2) = 12. P(G, −1) = (−1)4 − 4(−1)3 + 5(−1)2 − 2(−1)

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O = which is acyclic. # of acyclic orientations of G = (# for the triangle)(# for the remaining edge) = (23 − 2)(2) = 12. P(G, −1) = (−1)4 − 4(−1)3 + 5(−1)2 − 2(−1) = 12.

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O = which is acyclic. # of acyclic orientations of G = (# for the triangle)(# for the remaining edge) = (23 − 2)(2) = 12. P(G, −1) = (−1)4 − 4(−1)3 + 5(−1)2 − 2(−1) = 12.

Theorem (Stanley, 1973)

For any graph G with |V | = n, P(G, −1) = (−1)n(# of acyclic orientations of G).

An orientation of graph G = (V , E) is a directed graph O obtained by replacing each uv ∈ E by one of the arcs uv or

• vu. So the

number of orientations of G is 2|E|. A directed cycle of O is a sequence of distinct vertices v1, v2, . . . , vk with

• vivi+1 an arc for all

i modulo k. Orientation O is acyclic if it has no directed cycles. Ex. G = has orientation O = which is acyclic. # of acyclic orientations of G = (# for the triangle)(# for the remaining edge) = (23 − 2)(2) = 12. P(G, −1) = (−1)4 − 4(−1)3 + 5(−1)2 − 2(−1) = 12.

Theorem (Stanley, 1973)

For any graph G with |V | = n, P(G, −1) = (−1)n(# of acyclic orientations of G). Note: Blass and S (1986) gave a bijective proof of this theorem.

Richard P. Stanley Andreas Blass

A hyperplane in Rn is a subspace H with dim H = n − 1.

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}.

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

y = −x y = 2x

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

y = −x y = 2x

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x Let [n] = {1, 2, . . . , n}.

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x Let [n] = {1, 2, . . . , n}. Graph G with V = [n] has arrangement A(G) = {xi = xj : ij ∈ E}.

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x Let [n] = {1, 2, . . . , n}. Graph G with V = [n] has arrangement A(G) = {xi = xj : ij ∈ E}. Ex. G = 2 3 1

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x Let [n] = {1, 2, . . . , n}. Graph G with V = [n] has arrangement A(G) = {xi = xj : ij ∈ E}. Ex. G = 2 3 1 A(G) = {x1 = x2, x1 = x3} ⊂ R3.

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x Let [n] = {1, 2, . . . , n}. Graph G with V = [n] has arrangement A(G) = {xi = xj : ij ∈ E}. Ex. G = 2 3 1 A(G) = {x1 = x2, x1 = x3} ⊂ R3. # of regions of A(G) = 4.

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x Let [n] = {1, 2, . . . , n}. Graph G with V = [n] has arrangement A(G) = {xi = xj : ij ∈ E}. Ex. G = 2 3 1 A(G) = {x1 = x2, x1 = x3} ⊂ R3. # of regions of A(G) = 4. P(G, t) = t(t − 1)2

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x Let [n] = {1, 2, . . . , n}. Graph G with V = [n] has arrangement A(G) = {xi = xj : ij ∈ E}. Ex. G = 2 3 1 A(G) = {x1 = x2, x1 = x3} ⊂ R3. # of regions of A(G) = 4. P(G, t) = t(t − 1)2 = ⇒ P(G, −1) = −(−2)2 = −4.

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x Let [n] = {1, 2, . . . , n}. Graph G with V = [n] has arrangement A(G) = {xi = xj : ij ∈ E}. Ex. G = 2 3 1 A(G) = {x1 = x2, x1 = x3} ⊂ R3. # of regions of A(G) = 4. P(G, t) = t(t − 1)2 = ⇒ P(G, −1) = −(−2)2 = −4.

Theorem (Zaslavsky, 1975)

For any graph G with V = [n], P(G, −1) = (−1)n(# of regions of A(G)).

A hyperplane in Rn is a subspace H with dim H = n − 1. A hyperplane arrangement is a set of hyperplanes A = {H1, . . . , Hk}. The regions of A are the connected components of Rn − ∪iHi.

• Ex. A = {y = 2x, y = −x} ⊂ R2.

# of regions of A = 4. y = −x y = 2x Let [n] = {1, 2, . . . , n}. Graph G with V = [n] has arrangement A(G) = {xi = xj : ij ∈ E}. Ex. G = 2 3 1 A(G) = {x1 = x2, x1 = x3} ⊂ R3. # of regions of A(G) = 4. P(G, t) = t(t − 1)2 = ⇒ P(G, −1) = −(−2)2 = −4.

Theorem (Zaslavsky, 1975)

For any graph G with V = [n], P(G, −1) = (−1)n(# of regions of A(G)). There is a bijection: acyclic orientations of G ↔ regions of A(G).

Thomas Zaslavsky

Outline

Initial definitions The chromatic polynomial Acyclic orientations and hyperplane arrangements Increasing forests Comments and open questions

A graph F is forest if it has no (undirected) cycles.

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G).

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G). Let G be a graph with

V = [n] and F be a spanning forest.

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G). Let G be a graph with

V = [n] and F be a spanning forest. Ex. 4 3 1 2 G

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G). Let G be a graph with

V = [n] and F be a spanning forest. Ex. 4 3 1 2 G 4 3 1 2 F

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G). Let G be a graph with

V = [n] and F be a spanning forest. Then F is increasing if the vertices on any path of F starting at the minimum vertex of its connected component form an increasing sequence. Ex. 4 3 1 2 G 4 3 1 2 F

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G). Let G be a graph with

V = [n] and F be a spanning forest. Then F is increasing if the vertices on any path of F starting at the minimum vertex of its connected component form an increasing sequence. Ex. 4 3 1 2 G 4 3 1 2 F increasing

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G). Let G be a graph with

V = [n] and F be a spanning forest. Then F is increasing if the vertices on any path of F starting at the minimum vertex of its connected component form an increasing sequence. Ex. 4 3 1 2 G 4 3 1 2 F increasing 4 3 1 2

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G). Let G be a graph with

V = [n] and F be a spanning forest. Then F is increasing if the vertices on any path of F starting at the minimum vertex of its connected component form an increasing sequence. Ex. 4 3 1 2 G 4 3 1 2 F increasing 4 3 1 2 F not increasing

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G). Let G be a graph with

V = [n] and F be a spanning forest. Then F is increasing if the vertices on any path of F starting at the minimum vertex of its connected component form an increasing sequence. Ex. 4 3 1 2 G 4 3 1 2 F increasing 4 3 1 2 F not increasing Define isfm(G) = # of increasing spanning forests of G with m edges.

A graph F is forest if it has no (undirected) cycles. A subgraph H

• f a graph G is spanning if V (H) = V (G). Let G be a graph with

V = [n] and F be a spanning forest. Then F is increasing if the vertices on any path of F starting at the minimum vertex of its connected component form an increasing sequence. Ex. 4 3 1 2 G 4 3 1 2 F increasing 4 3 1 2 F not increasing Define isfm(G) = # of increasing spanning forests of G with m edges. and ISF(G, t) =

n−1

• m=0

(−1)m isfm(G)tn−m.

isfm(G) = # of increasing spanning forests of G with m edges. ISF(G) = ISF(G, t) =

• m≥0

(−1)m isfm(G)tn−m.

isfm(G) = # of increasing spanning forests of G with m edges. ISF(G) = ISF(G, t) =

• m≥0

(−1)m isfm(G)tn−m. 4 3 1 2 G = Ex.

isfm(G) = # of increasing spanning forests of G with m edges. ISF(G) = ISF(G, t) =

• m≥0

(−1)m isfm(G)tn−m. 4 3 1 2 G = Ex. isf0(G) = 1

isfm(G) = # of increasing spanning forests of G with m edges. ISF(G) = ISF(G, t) =

• m≥0

(−1)m isfm(G)tn−m. 4 3 1 2 G = Ex. isf0(G) = 1 isf1(G) = |E| = 4

isfm(G) = # of increasing spanning forests of G with m edges. ISF(G) = ISF(G, t) =

• m≥0

(−1)m isfm(G)tn−m. 4 3 1 2 G = Ex. 4 3 1 2 not increasing: isf0(G) = 1 isf1(G) = |E| = 4 isf2(G) = 4 2

• − 1 = 5

isfm(G) = # of increasing spanning forests of G with m edges. ISF(G) = ISF(G, t) =

• m≥0

(−1)m isfm(G)tn−m. 4 3 1 2 G = Ex. 4 3 1 2 not increasing: isf0(G) = 1 isf1(G) = |E| = 4 isf2(G) = 4 2

• − 1 = 5

isf3(G) = 4 3

• − 2 = 2

isfm(G) = # of increasing spanning forests of G with m edges. ISF(G) = ISF(G, t) =

• m≥0

(−1)m isfm(G)tn−m. 4 3 1 2 G = Ex. 4 3 1 2 not increasing: isf0(G) = 1 isf1(G) = |E| = 4 isf2(G) = 4 2

• − 1 = 5

isf3(G) = 4 3

• − 2 = 2

isf4(G) = 0

isfm(G) = # of increasing spanning forests of G with m edges. ISF(G) = ISF(G, t) =

• m≥0

(−1)m isfm(G)tn−m. 4 3 1 2 G = Ex. 4 3 1 2 not increasing: isf0(G) = 1 isf1(G) = |E| = 4 isf2(G) = 4 2

• − 1 = 5

isf3(G) = 4 3

• − 2 = 2

isf4(G) = 0 ISF(G) = t4 − 4t3 + 5t2 − 2t

isfm(G) = # of increasing spanning forests of G with m edges. ISF(G) = ISF(G, t) =

• m≥0

(−1)m isfm(G)tn−m. 4 3 1 2 G = Ex. 4 3 1 2 not increasing: isf0(G) = 1 isf1(G) = |E| = 4 isf2(G) = 4 2

• − 1 = 5

isf3(G) = 4 3

• − 2 = 2

isf4(G) = 0 ISF(G) = t4 − 4t3 + 5t2 − 2t = t(t − 1)2(t − 2).

Let G be a graph with vertex set V = [n].

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}.

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex.

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅,

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2 = {1} because 12 ∈ E,

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2 = {1} because 12 ∈ E, V3

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2 = {1} because 12 ∈ E, V3 = {2} because 23 ∈ E,

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2 = {1} because 12 ∈ E, V3 = {2} because 23 ∈ E, V4

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2 = {1} because 12 ∈ E, V3 = {2} because 23 ∈ E, V4 = {1, 2} because 14, 24 ∈ E

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2 = {1} because 12 ∈ E, V3 = {2} because 23 ∈ E, V4 = {1, 2} because 14, 24 ∈ E and (t −|V1|)(t −|V2|)(t −|V3|)(t −|V4|)

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2 = {1} because 12 ∈ E, V3 = {2} because 23 ∈ E, V4 = {1, 2} because 14, 24 ∈ E and (t −|V1|)(t −|V2|)(t −|V3|)(t −|V4|) = t(t −1)2(t −2)

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2 = {1} because 12 ∈ E, V3 = {2} because 23 ∈ E, V4 = {1, 2} because 14, 24 ∈ E and (t −|V1|)(t −|V2|)(t −|V3|)(t −|V4|) = t(t −1)2(t −2) = ISF(G).

Let G be a graph with vertex set V = [n]. For j ∈ [n] define Vj = {i ∈ V | i < j and ij ∈ E}. 4 3 1 2 G = Ex. V1 = ∅, V2 = {1} because 12 ∈ E, V3 = {2} because 23 ∈ E, V4 = {1, 2} because 14, 24 ∈ E and (t −|V1|)(t −|V2|)(t −|V3|)(t −|V4|) = t(t −1)2(t −2) = ISF(G).

Theorem (Hallam-S, 2014)

Let G have V = [n] and Vj as defined above. Then ISF(G, t) =

n

• j=1

(t − |Vj|).

Joshua Hallam

Vj = {i ∈ V | i < j and ij ∈ E}.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)?

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W .

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not G[V1] = G[∅]

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not ∅ G[V1] = G[∅]

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not ∅ G[V1] = G[∅] G[V2] = G[1]

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not ∅ G[V1] = G[∅] 1 G[V2] = G[1]

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not ∅ G[V1] = G[∅] 1 G[V2] = G[1] G[V3] = G[2]

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not ∅ G[V1] = G[∅] 1 G[V2] = G[1] 2 G[V3] = G[2]

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not ∅ G[V1] = G[∅] 1 G[V2] = G[1] 2 G[V3] = G[2] G[V4] = G[1, 2]

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not ∅ G[V1] = G[∅] 1 G[V2] = G[1] 2 G[V3] = G[2] 1 2 G[V4] = G[1, 2]

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

Vj = {i ∈ V | i < j and ij ∈ E}. When is ISF(G, t) = P(G, t)? If G is a graph and W ⊆ V (G), let G[W ] be the induced subgraph of G obtained by removing the vertices not in W . A clique is a graph H where every pair of vertices is connected by an edge. Ex. G = 4 3 1 2 G[1, 2, 4] = 4 1 2 clique G[2, 3, 4] = 4 3 2 not ∅ G[V1] = G[∅] 1 G[V2] = G[1] 2 G[V3] = G[2] 1 2 G[V4] = G[1, 2]

Theorem (Hallam-S, 2014)

Let G be a graph with V = [n]. Then P(G, t) = ISF(G, t) if and

• nly if the graphs G[Vj] are cliques for all 1 ≤ j ≤ n.

This clique condition is called a perfect elimination order.

Outline

Initial definitions The chromatic polynomial Acyclic orientations and hyperplane arrangements Increasing forests Comments and open questions

• 1. P(G, t) for negative integers.

• 1. P(G, t) for negative integers. Let G be a graph with acyclic
• rientation O.

• 1. P(G, t) for negative integers. Let G be a graph with acyclic
• rientation O. Let t be a positive integer and

c : V → {1, 2, . . . , t} be a coloring.

• 1. P(G, t) for negative integers. Let G be a graph with acyclic
• rientation O. Let t be a positive integer and

c : V → {1, 2, . . . , t} be a coloring. Call (O, c) compatible if

• uv ∈ O =

⇒ c(u) ≤ c(v).

• 1. P(G, t) for negative integers. Let G be a graph with acyclic
• rientation O. Let t be a positive integer and

c : V → {1, 2, . . . , t} be a coloring. Call (O, c) compatible if

• uv ∈ O =

⇒ c(u) ≤ c(v).

Theorem (Stanley, 1973)

Let G have |V | = n and let t be a positive integer. Then P(G, −t) = (−1)n(# of compatible pairs (O, c)).

• 1. P(G, t) for negative integers. Let G be a graph with acyclic
• rientation O. Let t be a positive integer and

c : V → {1, 2, . . . , t} be a coloring. Call (O, c) compatible if

• uv ∈ O =

⇒ c(u) ≤ c(v).

Theorem (Stanley, 1973)

Let G have |V | = n and let t be a positive integer. Then P(G, −t) = (−1)n(# of compatible pairs (O, c)). There is a bijective proof of this result due to S and Vatter (2019).

• 1. P(G, t) for negative integers. Let G be a graph with acyclic
• rientation O. Let t be a positive integer and

c : V → {1, 2, . . . , t} be a coloring. Call (O, c) compatible if

• uv ∈ O =

⇒ c(u) ≤ c(v).

Theorem (Stanley, 1973)

Let G have |V | = n and let t be a positive integer. Then P(G, −t) = (−1)n(# of compatible pairs (O, c)). There is a bijective proof of this result due to S and Vatter (2019).

• 2. Arbitrary hyperplanes.

• 1. P(G, t) for negative integers. Let G be a graph with acyclic
• rientation O. Let t be a positive integer and

c : V → {1, 2, . . . , t} be a coloring. Call (O, c) compatible if

• uv ∈ O =

⇒ c(u) ≤ c(v).

Theorem (Stanley, 1973)

Let G have |V | = n and let t be a positive integer. Then P(G, −t) = (−1)n(# of compatible pairs (O, c)). There is a bijective proof of this result due to S and Vatter (2019).

• 2. Arbitrary hyperplanes. Any hyperplane arrangement A has an

associated characteristic polynomial ch(A, t).

• 1. P(G, t) for negative integers. Let G be a graph with acyclic
• rientation O. Let t be a positive integer and

c : V → {1, 2, . . . , t} be a coloring. Call (O, c) compatible if

• uv ∈ O =

⇒ c(u) ≤ c(v).

Theorem (Stanley, 1973)

Let G have |V | = n and let t be a positive integer. Then P(G, −t) = (−1)n(# of compatible pairs (O, c)). There is a bijective proof of this result due to S and Vatter (2019).

• 2. Arbitrary hyperplanes. Any hyperplane arrangement A has an

associated characteristic polynomial ch(A, t). If A = A(G) then ch(A, t) = P(G, t).

• 1. P(G, t) for negative integers. Let G be a graph with acyclic
• rientation O. Let t be a positive integer and

c : V → {1, 2, . . . , t} be a coloring. Call (O, c) compatible if

• uv ∈ O =

⇒ c(u) ≤ c(v).

Theorem (Stanley, 1973)

Let G have |V | = n and let t be a positive integer. Then P(G, −t) = (−1)n(# of compatible pairs (O, c)). There is a bijective proof of this result due to S and Vatter (2019).

• 2. Arbitrary hyperplanes. Any hyperplane arrangement A has an

associated characteristic polynomial ch(A, t). If A = A(G) then ch(A, t) = P(G, t).

Theorem (Zaslavsky, 1975)

For any hyperplane arrangement A in Rn ch(A, −1) = (−1)n(# of regions of A).

• 3. Symmetric functions.

• 3. Symmetric functions. Consider variables x = {x1, x2, x3, . . . }.

• 3. Symmetric functions. Consider variables x = {x1, x2, x3, . . . }.

Power series f (x) is a symmetric function if it is invariant under permutations of x.

• 3. Symmetric functions. Consider variables x = {x1, x2, x3, . . . }.

Power series f (x) is a symmetric function if it is invariant under permutations of x. Bases for the vector space of symmetric functions homogeneous of degree n are indexed by partitions λ = (λ1, . . . , λk) of n.

• 3. Symmetric functions. Consider variables x = {x1, x2, x3, . . . }.

Power series f (x) is a symmetric function if it is invariant under permutations of x. Bases for the vector space of symmetric functions homogeneous of degree n are indexed by partitions λ = (λ1, . . . , λk) of n. The length of λ, ℓ(λ), is the number of λi.

• 3. Symmetric functions. Consider variables x = {x1, x2, x3, . . . }.

Power series f (x) is a symmetric function if it is invariant under permutations of x. Bases for the vector space of symmetric functions homogeneous of degree n are indexed by partitions λ = (λ1, . . . , λk) of n. The length of λ, ℓ(λ), is the number of λi. The chromatic symmetric function of G with V = {v1, . . . , vn} is X(G) = X(G, x) =

• c

xc(v1) . . . xc(vn) where the sum is over all proper colorings c : V → Z+.

• 3. Symmetric functions. Consider variables x = {x1, x2, x3, . . . }.

Power series f (x) is a symmetric function if it is invariant under permutations of x. Bases for the vector space of symmetric functions homogeneous of degree n are indexed by partitions λ = (λ1, . . . , λk) of n. The length of λ, ℓ(λ), is the number of λi. The chromatic symmetric function of G with V = {v1, . . . , vn} is X(G) = X(G, x) =

• c

xc(v1) . . . xc(vn) where the sum is over all proper colorings c : V → Z+. Setting x1 = · · · = xt = 1 and xi = 0 for i > t gives X(G; x) = P(G; t).

• 3. Symmetric functions. Consider variables x = {x1, x2, x3, . . . }.

Power series f (x) is a symmetric function if it is invariant under permutations of x. Bases for the vector space of symmetric functions homogeneous of degree n are indexed by partitions λ = (λ1, . . . , λk) of n. The length of λ, ℓ(λ), is the number of λi. The chromatic symmetric function of G with V = {v1, . . . , vn} is X(G) = X(G, x) =

• c

xc(v1) . . . xc(vn) where the sum is over all proper colorings c : V → Z+. Setting x1 = · · · = xt = 1 and xi = 0 for i > t gives X(G; x) = P(G; t). A sink in a directed graph is a vertex v with no outgoing arcs.

• 3. Symmetric functions. Consider variables x = {x1, x2, x3, . . . }.

Power series f (x) is a symmetric function if it is invariant under permutations of x. Bases for the vector space of symmetric functions homogeneous of degree n are indexed by partitions λ = (λ1, . . . , λk) of n. The length of λ, ℓ(λ), is the number of λi. The chromatic symmetric function of G with V = {v1, . . . , vn} is X(G) = X(G, x) =

• c

xc(v1) . . . xc(vn) where the sum is over all proper colorings c : V → Z+. Setting x1 = · · · = xt = 1 and xi = 0 for i > t gives X(G; x) = P(G; t). A sink in a directed graph is a vertex v with no outgoing arcs.

Theorem (Stanley, 1995)

Let eλ be the elementary symmetric function corresponding to λ. If we write X(G) =

λ cλeλ, then

# of acyclic orientations of G with k sinks =

• ℓ(λ)=k

cλ.

• 3. Symmetric functions. Consider variables x = {x1, x2, x3, . . . }.

Power series f (x) is a symmetric function if it is invariant under permutations of x. Bases for the vector space of symmetric functions homogeneous of degree n are indexed by partitions λ = (λ1, . . . , λk) of n. The length of λ, ℓ(λ), is the number of λi. The chromatic symmetric function of G with V = {v1, . . . , vn} is X(G) = X(G, x) =

• c

xc(v1) . . . xc(vn) where the sum is over all proper colorings c : V → Z+. Setting x1 = · · · = xt = 1 and xi = 0 for i > t gives X(G; x) = P(G; t). A sink in a directed graph is a vertex v with no outgoing arcs.

Theorem (Stanley, 1995)

Let eλ be the elementary symmetric function corresponding to λ. If we write X(G) =

λ cλeλ, then

# of acyclic orientations of G with k sinks =

• ℓ(λ)=k

cλ. While X(G) does not satisfy a DC Lemma, one does have such a result if the variables are noncommutative (Gebhard-S).

David Gebhard Jeremy Martin

• 4. More on increasing forests.

• 4. More on increasing forests. Hallam, Martin, and S (2018)

have extended these results to simplicial complexes of arbitrary dimension d.

• 4. More on increasing forests. Hallam, Martin, and S (2018)

have extended these results to simplicial complexes of arbitrary dimension d. When d = 1 one recovers the theorems for graphs.

• 4. More on increasing forests. Hallam, Martin, and S (2018)

have extended these results to simplicial complexes of arbitrary dimension d. When d = 1 one recovers the theorems for graphs. An increasing sequence is just one which avoids the pattern 21.

• 4. More on increasing forests. Hallam, Martin, and S (2018)

have extended these results to simplicial complexes of arbitrary dimension d. When d = 1 one recovers the theorems for graphs. An increasing sequence is just one which avoids the pattern 21. What can be said about spanning forests satisfying other pattern avoidance conditions?

• 4. More on increasing forests. Hallam, Martin, and S (2018)

have extended these results to simplicial complexes of arbitrary dimension d. When d = 1 one recovers the theorems for graphs. An increasing sequence is just one which avoids the pattern 21. What can be said about spanning forests satisfying other pattern avoidance conditions?

• 5. Log concavity.

• 4. More on increasing forests. Hallam, Martin, and S (2018)

have extended these results to simplicial complexes of arbitrary dimension d. When d = 1 one recovers the theorems for graphs. An increasing sequence is just one which avoids the pattern 21. What can be said about spanning forests satisfying other pattern avoidance conditions?

• 5. Log concavity. A polyomial P(t) = a0 + a1t + · · · + antn is log

concave if a2

k ≥ ak−1ak+1

for all 0 < k < n.

• 4. More on increasing forests. Hallam, Martin, and S (2018)

have extended these results to simplicial complexes of arbitrary dimension d. When d = 1 one recovers the theorems for graphs. An increasing sequence is just one which avoids the pattern 21. What can be said about spanning forests satisfying other pattern avoidance conditions?

• 5. Log concavity. A polyomial P(t) = a0 + a1t + · · · + antn is log

concave if a2

k ≥ ak−1ak+1

for all 0 < k < n. Using deep methods from algebraic geometry (Chern classes, etc.), Huh has proven the following.

• 4. More on increasing forests. Hallam, Martin, and S (2018)

have extended these results to simplicial complexes of arbitrary dimension d. When d = 1 one recovers the theorems for graphs. An increasing sequence is just one which avoids the pattern 21. What can be said about spanning forests satisfying other pattern avoidance conditions?

• 5. Log concavity. A polyomial P(t) = a0 + a1t + · · · + antn is log

concave if a2

k ≥ ak−1ak+1

for all 0 < k < n. Using deep methods from algebraic geometry (Chern classes, etc.), Huh has proven the following.

Theorem (Huh, 2013)

For any graph G, the polynomial P(G, t) is log concave.

• 4. More on increasing forests. Hallam, Martin, and S (2018)

have extended these results to simplicial complexes of arbitrary dimension d. When d = 1 one recovers the theorems for graphs. An increasing sequence is just one which avoids the pattern 21. What can be said about spanning forests satisfying other pattern avoidance conditions?

• 5. Log concavity. A polyomial P(t) = a0 + a1t + · · · + antn is log

concave if a2

k ≥ ak−1ak+1

for all 0 < k < n. Using deep methods from algebraic geometry (Chern classes, etc.), Huh has proven the following.

Theorem (Huh, 2013)

For any graph G, the polynomial P(G, t) is log concave. Recently (2015) Adiprasito, Huh, and Katz gave a combinatorial Hodge Theory proof of the Heron-Rota-Welsh conjecture which generalizes Huh’s result to any matroid.

Karim Adiprasito June Huh Eric Katz

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