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Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Polynomial graph and matroid invariants from graph homomorphisms

Delia Garijo1 Andrew Goodall2 Patrice Ossona de Mendez3 Jarik Neˇ setˇ ril2 Guus Regts4 and Llu´ ıs Vena2

1University of Seville 2Charles University, Prague 3CAMS, CNRS/EHESS, Paris 4University of Amsterdam

14 June 2016 Schloss Dagstuhl

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Graph polynomials Graph homomorphisms

Chromatic polynomial

Definition (Evaluation at positive integers) k ∈ N, P(G; k) = #{proper vertex k-colourings of G}.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Graph polynomials Graph homomorphisms

Chromatic polynomial

Definition (Evaluation at positive integers) k ∈ N, P(G; k) = #{proper vertex k-colourings of G}. e ∈ E(G) : P(G; k) = P(G\e; k) − P(G/e; k)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Graph polynomials Graph homomorphisms

Flow polynomial

Definition (Evaluation at positive integers) k ∈ N, F(G; k) = #{nowhere-zero Zk-flows of G}. e ∈ E(G) : F(G; k) =      F(G/e) − F(G\e) e ordinary e a bridge (k − 1)F(G\e) e a loop

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Graph polynomials Graph homomorphisms

Tutte polynomial

Definition (Subgraph expansion) For graph G = (V , E), T(G; x, y) =

• A⊆E

(x − 1)r(E)−r(A)(y − 1)|A|−r(A), where r(A) is the rank of the spanning subgraph (V , A) of G.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Graph polynomials Graph homomorphisms

Tutte polynomial

Definition (Subgraph expansion) For graph G = (V , E), T(G; x, y) =

• A⊆E

(x − 1)r(E)−r(A)(y − 1)|A|−r(A), where r(A) is the rank of the spanning subgraph (V , A) of G. T(G; x, y) =      T(G/e; x, y) + T(G\e; x, y) e ordinary xT(G/e; x, y) e a bridge yT(G\e; x, y) e a loop, and T(G; x, y) = 1 if G has no edges.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Graph polynomials Graph homomorphisms

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Graph polynomials Graph homomorphisms

Definition Graphs G, H. f : V (G) → V (H) is a homomorphism from G to H if uv ∈ E(G) ⇒ f (u)f (v) ∈ E(H).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Graph polynomials Graph homomorphisms

Definition Graphs G, H. f : V (G) → V (H) is a homomorphism from G to H if uv ∈ E(G) ⇒ f (u)f (v) ∈ E(H). Definition H with adjacency matrix (as,t), weight as,t on st ∈ E(H), hom(G, H) =

• f :V (G)→V (H)
• uv∈E(G)

af (u),f (v).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Graph polynomials Graph homomorphisms

Definition Graphs G, H. f : V (G) → V (H) is a homomorphism from G to H if uv ∈ E(G) ⇒ f (u)f (v) ∈ E(H). Definition H with adjacency matrix (as,t), weight as,t on st ∈ E(H), hom(G, H) =

• f :V (G)→V (H)
• uv∈E(G)

af (u),f (v). hom(G, H) = #{homomorphisms from G to H} = #{H-colourings of G} when H simple (as,t ∈ {0, 1}) or multigraph (as,t ∈ N)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 1

b b bc b bc b b bc bc

b

(Kk)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 1

b b bc b bc b b bc bc

b

(Kk) hom(G, Kk) = P(G; k) chromatic polynomial

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Problem Which sequences (Hk) of graphs are such that, for all graphs G, for each k ∈ N we have hom(G, Hk) = p(G; k) for a fixed polynomial p(G)?

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Problem Which sequences (Hk) of graphs are such that, for all graphs G, for each k ∈ N we have hom(G, Hk) = p(G; k) for a fixed polynomial p(G)? Example For all graphs G, hom(G, Kk) = P(G; k) is the evaluation of the chromatic polynomial of G at k.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 2: add loops

b b bc b bc b b bc bc

b

(K 1

k )

hom(G, K 1

k ) = k|V (G)|

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 3: add ℓ loops

b b bc b bc b b bc bc

b

ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ

(K ℓ

k)

hom(G, K ℓ

k) =

• f :V (G)→[k]

ℓ#{uv∈E(G) | f (u)=f (v)}

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 3: add ℓ loops

b b bc b bc b b bc bc

b

ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ

(K ℓ

k)

hom(G, K ℓ

k) =

• f :V (G)→[k]

ℓ#{uv∈E(G) | f (u)=f (v)} = kc(G)(ℓ − 1)r(G)T(G; ℓ−1+k

ℓ−1 , ℓ) (Tutte polynomial)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 4: add loops weight 1 − k

b b bc b bc b b bc bc

b

−1 −1 −2 −2 −2 −3 −3 −3 −3

(K 1−k

k

) hom(G, K 1−k

k

) =

• f :V (G)→[k]

(1 − k)#{uv∈E(G) | f (u)=f (v)}

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 4: add loops weight 1 − k

b b bc b bc b b bc bc

b

−1 −1 −2 −2 −2 −3 −3 −3 −3

(K 1−k

k

) hom(G, K 1−k

k

) =

• f :V (G)→[k]

(1 − k)#{uv∈E(G) | f (u)=f (v)} = (−1)|E(G)|k|V (G)|F(G; k) (flow polynomial)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 5

b bc

b bc b bc b bc bc b b bc b bc

(K k

2 ) = (Qk) (hypercubes)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 5

b bc

b bc b bc b bc bc b b bc b bc

(K k

2 ) = (Qk) (hypercubes)

Proposition (Garijo, G., Neˇ setˇ ril, 2015) hom(G, Qk) = p(G; k, 2k) for bivariate polynomial p(G)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example 6

b b bc b bc b b bc bc

b

(Ck) hom(C3, C3) = 6, hom(C3, Ck) = 0 when k = 2 or k ≥ 4

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Definition (Hk) is strongly polynomial (in k) if ∀G ∃ polynomial p(G) such that hom(G, Hk) = p(G; k) for all k ∈ N.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Definition (Hk) is strongly polynomial (in k) if ∀G ∃ polynomial p(G) such that hom(G, Hk) = p(G; k) for all k ∈ N. Example (Kk), (K 1

k ) are strongly polynomial

(K ℓ

k) is strongly polynomial (in k, ℓ)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Definition (Hk) is strongly polynomial (in k) if ∀G ∃ polynomial p(G) such that hom(G, Hk) = p(G; k) for all k ∈ N. Example (Kk), (K 1

k ) are strongly polynomial

(K ℓ

k) is strongly polynomial (in k, ℓ)

(Qk) not strongly polynomial (but polynomial in k and 2k)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Definition (Hk) is strongly polynomial (in k) if ∀G ∃ polynomial p(G) such that hom(G, Hk) = p(G; k) for all k ∈ N. Example (Kk), (K 1

k ) are strongly polynomial

(K ℓ

k) is strongly polynomial (in k, ℓ)

(Qk) not strongly polynomial (but polynomial in k and 2k) (Ck), (Pk) not strongly polynomial (but eventually polynomial in k)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

De la Harpe & Jaeger (1995) construct families of strongly polynomial sequences, extended by Garijo, G. & Neˇ setˇ ril (2015).

Construction [G., Neˇ setˇ ril, Ossona de Mendez, 2016] Strongly polynomial sequences by quantifier-free (QF) interpretation of sequences of finite relational structures.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Constructing cycles

b bc b bc b b bc bc b

edge uv when

b b bc b bc b bc bc b bc bc bc

(u → v) ∧ ¬∃w(u → v ∧ w → v) ∨ ∀w(u → w ∧ w → v)

• r the same formula

with u and v swapped

Cycles (Ck) from tournaments Tk require quantification. Sequence (Ck) is not strongly polynomial.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Conjecture (G., Neˇ setˇ ril, Ossona de Mendez, 2016) All strongly polynomial sequences of graphs (Hk) such that Hk ⊆ind Hk+1 can be obtained by QF interpretation of a ”basic sequence” (disjoint union of transitive tournaments of size polynomial in k with unary relations).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Conjecture (G., Neˇ setˇ ril, Ossona de Mendez, 2016) All strongly polynomial sequences of graphs (Hk) such that Hk ⊆ind Hk+1 can be obtained by QF interpretation of a ”basic sequence” (disjoint union of transitive tournaments of size polynomial in k with unary relations). Theorem (G., Neˇ

setˇ ril, Ossona de Mendez , 2016)

A sequence (Hk) of graphs of uniformly bounded degree is a strongly polynomial sequence if and only if it is a QF-interpretation

• f a basic sequence.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Proposition Let (Hk) be a strongly polynomial sequence of graphs. For a graph G, let φ(G) be a quantifier-free formula in the language of graphs, whose free variables are indexed by vertices of G. Let pφ(G; k) equal the number satisfying assignments for φ(G) taking values in V (Hk). Then pφ(G; k) is a polynomial in k. Example A satisfying assignment (v1, . . . , vn) ∈ V (Hk)n of φ(G) =

• ij∈E(G)

(vi ∼ vj), corresponds precisely to a homomorphism i → vi. So here pφ(G; k) = p(G; k) = hom(G, Hk).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Example Take a strongly polynomial sequence (Hk) (such as (Kk), (K 1

k ),

half graphs, ...). The number of Hk-colourings of G for which (1) the monochromatic components belong to some fixed family

• f isomorphism types of graph (cocolourings, convex

colourings, ...) (2) the inverse image of any pair of colours induces a given type

• f graph (acyclic colourings, star colourings, harmonious

colourings, ...) is a polynomial function of k.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems Examples Strongly polynomial sequences of graphs

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Chromatic polynomial as a cycle matroid invariant

Proposition The graph invariant P(G; k) kc(G) = hom(G, Kk) kc(G) depends just on the cycle matroid of G: it counts the number of nowhere-zero Zk-tensions of G.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Cayley graphs

Cayley(A, B) denotes the Cayley graph on vertex set additive group A and B = −B a subset of A determining that edges join u and v precisely when u − v ∈ B.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Cayley graphs

Cayley(A, B) denotes the Cayley graph on vertex set additive group A and B = −B a subset of A determining that edges join u and v precisely when u − v ∈ B. Cayley( Zk, Zk \ {0} ) ∼ = Kk.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Cayley graphs

Cayley(A, B) denotes the Cayley graph on vertex set additive group A and B = −B a subset of A determining that edges join u and v precisely when u − v ∈ B. Cayley( Zk, Zk \ {0} ) ∼ = Kk. Proposition For abelian additive group A and B = −B a subset of A, the graph invariant G → hom(G, Cayley(A, B) ) |A|c(G) is a cycle matroid invariant and counts the number of B-tensions

• f G.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Definition Generalized Johnson graph Jk,r,D, D ⊆ {0, 1, . . . , r} vertices [k]

r

• ,

edge uv when |u ∩ v| ∈ D

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Definition Generalized Johnson graph Jk,r,D, D ⊆ {0, 1, . . . , r} vertices [k]

r

• ,

edge uv when |u ∩ v| ∈ D Johnson graphs D = {k − 1} J(k, r) Kneser graphs D = {0} Kk:r

In general not Cayley graphs.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Definition Generalized Johnson graph Jk,r,D, D ⊆ {0, 1, . . . , r} vertices [k]

r

• ,

edge uv when |u ∩ v| ∈ D Johnson graphs D = {k − 1} J(k, r) Kneser graphs D = {0} Kk:r

In general not Cayley graphs.

Petersen graph = K5:2

Figure by Watchduck (a.k.a. Tilman Piesk). Wikimedia Commons

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Proposition For a graph G and k, r ≥ 1, hom(G, Kk:r) = (r!)−|V (G)|P(G[Kr]; k).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Proposition For a graph G and k, r ≥ 1, hom(G, Kk:r) = (r!)−|V (G)|P(G[Kr]; k). Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Neˇ

setˇ ril, 2015)

For every r, D, sequence (Jk,r,D) is strongly polynomial (in k).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Proposition For a graph G and k, r ≥ 1, hom(G, Kk:r) = (r!)−|V (G)|P(G[Kr]; k). Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Neˇ

setˇ ril, 2015)

For every r, D, sequence (Jk,r,D) is strongly polynomial (in k). Proposition (de la Harpe & Jaeger, 1995) The graph parameter k

r

−c(G)hom(G, Jk,r,D) depends only on the cycle matroid of G.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Proposition For a graph G and k, r ≥ 1, hom(G, Kk:r) = (r!)−|V (G)|P(G[Kr]; k). Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Neˇ

setˇ ril, 2015)

For every r, D, sequence (Jk,r,D) is strongly polynomial (in k). Proposition (de la Harpe & Jaeger, 1995) The graph parameter k

r

−c(G)hom(G, Jk,r,D) depends only on the cycle matroid of G. Problem Interpret k

r

−c(G)hom(G, Jk,r,D) in terms of the cycle matroid of G alone.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Definition The action of a group Γ on a set S is transitive if for each s, t ∈ S there is γ ∈ Γ such that sγ = t. The action of a group Γ on a set S is generously transitive if for each s, t ∈ S there is γ ∈ Γ such that sγ = t and s = tγ.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Definition The action of a group Γ on a set S is transitive if for each s, t ∈ S there is γ ∈ Γ such that sγ = t. The action of a group Γ on a set S is generously transitive if for each s, t ∈ S there is γ ∈ Γ such that sγ = t and s = tγ. Theorem (de la Harpe & Jaeger, 1995) The graph invariant G → hom(G, H) |V (H)|c(G) depends just on the cycle matroid of G if H has generously transitive automorphism group.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Definition The action of a group Γ on a set S is transitive if for each s, t ∈ S there is γ ∈ Γ such that sγ = t. The action of a group Γ on a set S is generously transitive if for each s, t ∈ S there is γ ∈ Γ such that sγ = t and s = tγ. Theorem (G., Regts & Vena, 2016) The graph invariant G → hom(G, H) |V (H)|c(G) depends just on the cycle matroid of G only if H has generously transitive automorphism group.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Duality

For a planar graph G, T(G ∗; x, y) = T(G; y, x).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Duality

For a planar graph G, T(G ∗; x, y) = T(G; y, x). For a graph G, #{nowhere-zero Zk-tensions} = k−c(G)P(G; k) = (− 1)r(G)T(G; 1 − k, 0), #{nowhere-zero Zk-flows} = F(G; k) = (−1)n(G)T(G; 0, 1 − k).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Duality

For a planar graph G, T(G ∗; x, y) = T(G; y, x). For a graph G, #{nowhere-zero Zk-tensions} = k−c(G)P(G; k) = (− 1)r(G)T(G; 1 − k, 0), #{nowhere-zero Zk-flows} = F(G; k) = (−1)n(G)T(G; 0, 1 − k). Tutte polynomial extends to any matroid M = (E, r) defined

• n 2E by size | | and rank function r (or rank/nullity).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Duality

For a planar graph G, T(G ∗; x, y) = T(G; y, x). For a graph G, #{nowhere-zero Zk-tensions} = k−c(G)P(G; k) = (− 1)r(G)T(G; 1 − k, 0), #{nowhere-zero Zk-flows} = F(G; k) = (−1)n(G)T(G; 0, 1 − k). Tutte polynomial extends to any matroid M = (E, r) defined

• n 2E by size | | and rank function r (or rank/nullity).

Tensions/flows defined for orientable matroids.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

chromatic polynomial cycle matroid of a graph signed circuits (from edge orientations) nowhere-zero tensions cycle matroid of a graph signed cutsets (from edge orientation) nowhere-zero flows

• rientable matroid

signed circuits/ cocircuits nowhere-zero tensions/flows Tutte polynomial matroid rank/nullity

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

chromatic polynomial cycle matroid of a graph signed circuits (from edge orientations) nowhere-zero tensions cycle matroid of a graph signed cutsets (from edge orientation) nowhere-zero flows

• rientable matroid

signed circuits/ cocircuits nowhere-zero tensions/flows Tutte polynomial matroid rank/nullity P(G; k) = kc(G)(−1)r(G)T(G; 1 − k, 0)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

chromatic polynomial cycle matroid of a graph signed circuits (from edge orientations) nowhere-zero tensions cycle matroid of a graph signed cutsets (from edge orientation) nowhere-zero flows

• rientable matroid

signed circuits/ cocircuits nowhere-zero tensions/flows Tutte polynomial matroid rank/nullity P(G; k) = kc(G)(−1)r(G)T(G; 1 − k, 0) F(G; k) = (−1)|E|−r(G)T(G; 0, 1 − k)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

chromatic polynomial cycle matroid of a graph signed circuits (from edge orientations) nowhere-zero tensions cycle matroid of a graph signed cutsets (from edge orientation) nowhere-zero flows

• rientable matroid

signed circuits/ cocircuits nowhere-zero tensions/flows Tutte polynomial matroid rank/nullity P(G; k) = kc(G)(−1)r(G)T(G; 1 − k, 0) F(G; k) = (−1)|E|−r(G)T(G; 0, 1 − k) T(M; x, y) =

A⊆E(x−1)r(E)−r(A)(y−1)|A|−r(A)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

chromatic polynomial cycle matroid of a graph signed circuits (from edge orientations) nowhere-zero tensions cycle matroid of a graph signed cutsets (from edge orientation) nowhere-zero flows

• rientable matroid

signed circuits/ cocircuits nowhere-zero tensions/flows Tutte polynomial matroid rank/nullity P(G; k) = kc(G)(−1)r(G)T(G; 1 − k, 0) F(G; k) = (−1)|E|−r(G)T(G; 0, 1 − k) T(M; x, y) =

A⊆E(x−1)r(E)−r(A)(y−1)|A|−r(A)

common generalization to orientable matroids?

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

chromatic polynomial cycle matroid of a graph signed circuits (from edge orientations) nowhere-zero tensions cycle matroid of a graph signed cutsets (from edge orientation) nowhere-zero flows

• rientable matroid

signed circuits/ cocircuits nowhere-zero tensions/flows Tutte polynomial matroid rank/nullity proper vertex colourings P(G; k) k−c(G)P(G; k) F(G; k) ? T (M; x, y)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

chromatic polynomial cycle matroid of a graph signed circuits (from edge orientations) nowhere-zero tensions cycle matroid of a graph signed cutsets (from edge orientation) nowhere-zero flows

• rientable matroid

signed circuits/ cocircuits nowhere-zero tensions/flows Tutte polynomial matroid rank/nullity proper vertex colourings P(G; k) k−c(G)P(G; k) F(G; k) ? T (M; x, y) Kneser/Johnson colourings h

• m

( G , J

k , r , D

) a Kneser/Johnson Tutte polynomial? dual to Kneser/Johnson colourings?

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Abelian to non-Abelian, graphs to maps

cycle matroid of a graph circuit rotations (graph traversal) non-Abelian tensions? cycle matroid of a graph signed cutsets (from edge orientation) non-Abelian flows? “rotatable” orientable matroid? signed/rotated circuits/ cocircuits? nowhere-zero tensions/flows ? delta matroid? Kneser/Johnson colourings k

r

−c(G)hom(G, Jk,r,D) ? ? hom(G, Jk,r,D) signed circuits (from edge orientations) vertex rotations (orientable embedding)

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Vertex-transitive graphs are homomorphic images of Cayley graphs

• n finite groups (Sabidussi, 1964): finding hom(G, H) for

vertex-transitive H reduces to finding hom(G, H), where H is a Cayley graph with image H.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Vertex-transitive graphs are homomorphic images of Cayley graphs

• n finite groups (Sabidussi, 1964): finding hom(G, H) for

vertex-transitive H reduces to finding hom(G, H), where H is a Cayley graph with image H. Counting Kneser Kk:r-colourings reduces to counting Br-tensions for a certain inverse-closed subset Br of Sym(k).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Rational generating functions

◮ For strongly polynomial sequence (Hk),

• k

hom(G, Hk)tk = PG(t) (1 − t)|V (G)|+1 with polynomial PG(t) of degree at most |V (G)|.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Rational generating functions

◮ For strongly polynomial sequence (Hk),

• k

hom(G, Hk)tk = PG(t) (1 − t)|V (G)|+1 with polynomial PG(t) of degree at most |V (G)|.

◮ For eventually polynomial sequence (Hk) such as (Ck),

• k

hom(G, Hk)tk = PG(t) (1 − t)|V (G)|+1 with polynomial PG(t).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Beyond polynomials? Rational generating functions

◮ For quasipolynomial sequence of Tur´

an graphs (Tk,r)

• k

hom(G, Tk,r)tk = PG(t) Q(t)|V (G)|+1 with polynomial PG(t) of degree at most |V (G)| and polynomial Q(t) with zeros rth roots of unity.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Beyond polynomials? Rational generating functions

◮ For quasipolynomial sequence of Tur´

an graphs (Tk,r)

• k

hom(G, Tk,r)tk = PG(t) Q(t)|V (G)|+1 with polynomial PG(t) of degree at most |V (G)| and polynomial Q(t) with zeros rth roots of unity.

◮ For sequence of hypercubes (Qk),

• k

hom(G, Qk)tk = PG(t) Q(t)|V (G)|+1 with polynomial PG(t) of degree at most |V (G)| and polynomial Q(t) with zeros powers of 2.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Beyond polynomials? Algebraic generating functions

◮ For sequence of odd graphs Ok = J2k−1,k−1,{0}

• k

hom(G, Ok)tk is algebraic (e.g. 1

2(1 − 4t)− 1

2 when G = K1).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

◮ When is hom(G, Cayley(Ak, Bk)) a fixed polynomial

(dependent on G) in |Ak|, |Bk|, where Bk = −Bk ⊆ Ak?

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

◮ When is hom(G, Cayley(Ak, Bk)) a fixed polynomial

(dependent on G) in |Ak|, |Bk|, where Bk = −Bk ⊆ Ak?

(hypercubes) hom(G, Cayley(Zk

2, S1)) polynomial in 2k and k

(S1 = {weight 1 vectors}). [Garijo, G., Neˇ setˇ ril 2015]

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

◮ When is hom(G, Cayley(Ak, Bk)) a fixed polynomial

(dependent on G) in |Ak|, |Bk|, where Bk = −Bk ⊆ Ak?

(hypercubes) hom(G, Cayley(Zk

2, S1)) polynomial in 2k and k

(S1 = {weight 1 vectors}). [Garijo, G., Neˇ setˇ ril 2015] For D ⊂ N, hom(G, Cayley(Zk, ±D)) is polynomial in k for sufficiently large k iff D is finite or cofinite. [de la Harpe & Jaeger, 1995]

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

◮ When is hom(G, Cayley(Ak, Bk)) a fixed polynomial

(dependent on G) in |Ak|, |Bk|, where Bk = −Bk ⊆ Ak?

(hypercubes) hom(G, Cayley(Zk

2, S1)) polynomial in 2k and k

(S1 = {weight 1 vectors}). [Garijo, G., Neˇ setˇ ril 2015] For D ⊂ N, hom(G, Cayley(Zk, ±D)) is polynomial in k for sufficiently large k iff D is finite or cofinite. [de la Harpe & Jaeger, 1995] (circular colourings) hom(G, Cayley(Zks, {kr, kr +1, . . . , k(s−r)})) polynomial in

• k. [G., Neˇ

setˇ ril, Ossona de Mendez 2016]

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

◮ When is hom(G, Cayley(Ak, Bk)) a fixed polynomial

(dependent on G) in |Ak|, |Bk|, where Bk = −Bk ⊆ Ak?

(hypercubes) hom(G, Cayley(Zk

2, S1)) polynomial in 2k and k

(S1 = {weight 1 vectors}). [Garijo, G., Neˇ setˇ ril 2015] For D ⊂ N, hom(G, Cayley(Zk, ±D)) is polynomial in k for sufficiently large k iff D is finite or cofinite. [de la Harpe & Jaeger, 1995] (circular colourings) hom(G, Cayley(Zks, {kr, kr +1, . . . , k(s−r)})) polynomial in

• k. [G., Neˇ

setˇ ril, Ossona de Mendez 2016]

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

◮ When is hom(G, Cayley(Ak, Bk)) a fixed polynomial

(dependent on G) in |Ak|, |Bk|, where Bk = −Bk ⊆ Ak?

(hypercubes) hom(G, Cayley(Zk

2, S1)) polynomial in 2k and k

(S1 = {weight 1 vectors}). [Garijo, G., Neˇ setˇ ril 2015] For D ⊂ N, hom(G, Cayley(Zk, ±D)) is polynomial in k for sufficiently large k iff D is finite or cofinite. [de la Harpe & Jaeger, 1995] (circular colourings) hom(G, Cayley(Zks, {kr, kr +1, . . . , k(s−r)})) polynomial in

• k. [G., Neˇ

setˇ ril, Ossona de Mendez 2016]

◮ Which graph polynomials defined by strongly polynomial

sequences of graphs satisfy a reduction formula (size-decreasing recurrence) like the chromatic polynomial and independence polynomial?

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Four papers

• P. de la Harpe and F. Jaeger, Chromatic invariants for finite graphs:

theme and polynomial variations, Lin. Algebra Appl. 226–228 (1995), 687–722 Defining graphs invariants from counting graph homomorphisms.

• Examples. Basic constructions.
• D. Garijo, A. Goodall, J. Neˇ

setˇ ril, Polynomial graph invariants from homomorphism numbers. Discrete Math., 339 (2016), no. 4, 1315–1328. Early version at arXiv: 1308.3999 [math.CO] Further examples. New construction using rooted tree representations of graphs (e.g. cotrees).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Four papers

• A. Goodall, J. Neˇ

setˇ ril, P. Ossona de Mendez, Strongly polynomial sequences as interpretation of trivial structures. J. Appl. Logic, to

• appear. Also at arXiv:1405.2449 [math.CO].

General relational structures: counting satisfying assignments for quantifier-free formulas. Building new polynomial invariants by interpretation of ”trivial” sequences of marked tournaments. A.J. Goodall, G. Regts and L. Vena Cros, Matroid invariants and counting graph homomorphisms. Linear Algebra Appl. 494 (2016), 263–273. Preprint: arXiv:1512.01507 [math.CO] Cycle matroid invariants from counitng graph homomorphisms.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Prime power q = pd ≡ 1 (mod 4) Paley graph Pq =Cayley(Fq, non-zero squares),

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Prime power q = pd ≡ 1 (mod 4) Paley graph Pq =Cayley(Fq, non-zero squares), Quasi-random graphs: hom(G, Pq)/hom(G, Gq, 1

2 ) → 1 as q → ∞.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Prime power q = pd ≡ 1 (mod 4) Paley graph Pq =Cayley(Fq, non-zero squares), Quasi-random graphs: hom(G, Pq)/hom(G, Gq, 1

2 ) → 1 as q → ∞.

Proposition (Corollary to result of de la Harpe & Jaeger, 1995) hom(G, Pq) is polynomial in q for series-parallel G.

e.g. hom(K3, Pq) = q(q−1)(q−5)

8

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Prime power q = pd ≡ 1 (mod 4) Paley graph Pq =Cayley(Fq, non-zero squares), Quasi-random graphs: hom(G, Pq)/hom(G, Gq, 1

2 ) → 1 as q → ∞.

Proposition (Corollary to result of de la Harpe & Jaeger, 1995) hom(G, Pq) is polynomial in q for series-parallel G.

e.g. hom(K3, Pq) = q(q−1)(q−5)

8

Prime q ≡ 1 (mod 4), q = 4x2 + y2, [Evans, Pulham, Sheehan, 1981]: hom(K4, Pq) = q(q − 1) 1536

• (q − 9)2 − 4x2

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Prime power q = pd ≡ 1 (mod 4) Paley graph Pq =Cayley(Fq, non-zero squares), Quasi-random graphs: hom(G, Pq)/hom(G, Gq, 1

2 ) → 1 as q → ∞.

Proposition (Corollary to result of de la Harpe & Jaeger, 1995) hom(G, Pq) is polynomial in q for series-parallel G.

e.g. hom(K3, Pq) = q(q−1)(q−5)

8

Prime q ≡ 1 (mod 4), q = 4x2 + y2, [Evans, Pulham, Sheehan, 1981]: hom(K4, Pq) = q(q − 1) 1536

• (q − 9)2 − 4x2

Is hom(G, Pq) polynomial in q and x for all graphs G?

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B-tensions and B-flows to a B-Tutte? Beyond polynomials? Open problems

Prime power q = pd ≡ 1 (mod 4) Paley graph Pq =Cayley(Fq, non-zero squares), Quasi-random graphs: hom(G, Pq)/hom(G, Gq, 1

2 ) → 1 as q → ∞.

Proposition (Corollary to result of de la Harpe & Jaeger, 1995) hom(G, Pq) is polynomial in q for series-parallel G.

e.g. hom(K3, Pq) = q(q−1)(q−5)

8

Prime q ≡ 1 (mod 4), q = 4x2 + y2, [Evans, Pulham, Sheehan, 1981]: hom(K4, Pq) = q(q − 1) 1536

• (q − 9)2 − 4x2

Is hom(G, Pq) polynomial in q and x for all graphs G? Theorem (G., Neˇ

setˇ ril, Ossona de Mendez , 2014+)

If (Hk) is strongly polynomial then there are only finitely many terms belonging to a quasi-random sequence of graphs.