On Decomposition of Cartesian Products of Regular Graphs into - - PowerPoint PPT Presentation

On Decomposition of Cartesian Products of Regular Graphs into Isomorphic Trees

Kyle F. Jao

Department of Mathematics University of Illinois at Urbana-Champaign kylejao@gmail.com Joint work with

Alexandr V. Kostochka and Douglas B. West

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

The Problem

Let T be a fixed tree with m edges. A graph G has a T-decomposition if the edges of G can be partitioned so that each class forms a copy of T.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

The Problem

Let T be a fixed tree with m edges. A graph G has a T-decomposition if the edges of G can be partitioned so that each class forms a copy of T. Conjecture (Ringel [1964]) K2m+1 has a T-decomposition.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

The Problem

Let T be a fixed tree with m edges. A graph G has a T-decomposition if the edges of G can be partitioned so that each class forms a copy of T. Conjecture (Ringel [1964]) K2m+1 has a T-decomposition. Conjecture (Graham–H¨ aggkvist [1984]) Every 2m-regular graph has a T-decomposition. Every m-regular bipartite graph has a T-dcomposition.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Results

Snevily gave a short proof for Graham–H¨ aggkvist conjecture for the case girth(G) > diam(T).

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Results

Snevily gave a short proof for Graham–H¨ aggkvist conjecture for the case girth(G) > diam(T). Theorem (Snevily [1991]) Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with 1-fact.). If G has no cycle with length at most diam(T) consisting of edges in distinct F-classes, then G has a T-decomposition.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Results

Snevily gave a short proof for Graham–H¨ aggkvist conjecture for the case girth(G) > diam(T). Theorem (Snevily [1991]) Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with 1-fact.). If G has no cycle with length at most diam(T) consisting of edges in distinct F-classes, then G has a T-decomposition.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Results

Snevily gave a short proof for Graham–H¨ aggkvist conjecture for the case girth(G) > diam(T). Theorem (Snevily [1991]) Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with 1-fact.). If G has no cycle with length at most diam(T) consisting of edges in distinct F-classes, then G has a T-decomposition. Let G be the Cartesian product of G1, . . . , Gk, where Gi is a 2ri-regular graph with a 2-factorization Fi (or ri-reg. bip. with 1-fact.). Say r = (r1, . . . , rk) with r1 ≤ . . . ≤ rk and sum m.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Results

Snevily gave a short proof for Graham–H¨ aggkvist conjecture for the case girth(G) > diam(T). Theorem (Snevily [1991]) Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with 1-fact.). If G has no cycle with length at most diam(T) consisting of edges in distinct F-classes, then G has a T-decomposition. Let G be the Cartesian product of G1, . . . , Gk, where Gi is a 2ri-regular graph with a 2-factorization Fi (or ri-reg. bip. with 1-fact.). Say r = (r1, . . . , rk) with r1 ≤ . . . ≤ rk and sum m. An edge-coloring of T is r-exact if exactly ri edges have color i. Given an r-exact edge-coloring of T and establish a one-to-one correspondence between edges of color i in T and factors in Fi for each i.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Results

Snevily gave a short proof for Graham–H¨ aggkvist conjecture for the case girth(G) > diam(T). Theorem (Snevily [1991]) Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with 1-fact.). If G has no cycle with length at most diam(T) consisting of edges in distinct F-classes, then G has a T-decomposition. Let G be the Cartesian product of G1, . . . , Gk, where Gi is a 2ri-regular graph with a 2-factorization Fi (or ri-reg. bip. with 1-fact.). Say r = (r1, . . . , rk) with r1 ≤ . . . ≤ rk and sum m. An edge-coloring of T is r-exact if exactly ri edges have color i. Given an r-exact edge-coloring of T and establish a one-to-one correspondence between edges of color i in T and factors in Fi for each i. Theorem (J.–Kostochka–West [2011+]) If every path P in T uses a color i such that Gi has no cycle consisting of edges in distinct Fi-classes all corresponding to edges of P, then G has a T-decomposition.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Results

Snevily gave a short proof for Graham–H¨ aggkvist conjecture for the case girth(G) > diam(T). Theorem (Snevily [1991]) Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with 1-fact.). If G has no cycle with length at most diam(T) consisting of edges in distinct F-classes, then G has a T-decomposition. Let G be the Cartesian product of G1, . . . , Gk, where Gi is a 2ri-regular graph with a 2-factorization Fi (or ri-reg. bip. with 1-fact.). Say r = (r1, . . . , rk) with r1 ≤ . . . ≤ rk and sum m. An edge-coloring of T is r-exact if exactly ri edges have color i. Given an r-exact edge-coloring of T and establish a one-to-one correspondence between edges of color i in T and factors in Fi for each i. Theorem (J.–Kostochka–West [2011+]) If every path P in T uses a color i such that Gi has no cycle consisting of edges in distinct Fi-classes all corresponding to edges of P, then G has a T-decomposition.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Proof

Note that F1, . . . , Fk yield a 2-factorization of G by decomposing each copy of Gi according to Fi and combining these decompositions.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Proof

Note that F1, . . . , Fk yield a 2-factorization of G by decomposing each copy of Gi according to Fi and combining these decompositions. We prove a stronger result by induction on m. We produce a T-decomposition of G such that each vertex of G represents distinct vertices of T in m + 1 copies of T, and in each copy of T each edge e is embeded as an edge of the 2-factor corresponding to e.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Proof

Note that F1, . . . , Fk yield a 2-factorization of G by decomposing each copy of Gi according to Fi and combining these decompositions. We prove a stronger result by induction on m. We produce a T-decomposition of G such that each vertex of G represents distinct vertices of T in m + 1 copies of T, and in each copy of T each edge e is embeded as an edge of the 2-factor corresponding to e. For m = 1, okay. Consider m > 1, let u be a leaf with neighbor v and T ′ = T − u. May assume uv has color k and corresponds to the 2-factor H of Gk in Fk.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Proof

Note that F1, . . . , Fk yield a 2-factorization of G by decomposing each copy of Gi according to Fi and combining these decompositions. We prove a stronger result by induction on m. We produce a T-decomposition of G such that each vertex of G represents distinct vertices of T in m + 1 copies of T, and in each copy of T each edge e is embeded as an edge of the 2-factor corresponding to e. For m = 1, okay. Consider m > 1, let u be a leaf with neighbor v and T ′ = T − u. May assume uv has color k and corresponds to the 2-factor H of Gk in Fk. Let G ′ be the Cartesian product of G1, . . . , Gk−1, Gk − E(H). Consider the T ′-decomposition of G ′ provided by the induction hypothesis. Each vertex of G ′ represents v in some copy of T ′.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Proof

Note that F1, . . . , Fk yield a 2-factorization of G by decomposing each copy of Gi according to Fi and combining these decompositions. We prove a stronger result by induction on m. We produce a T-decomposition of G such that each vertex of G represents distinct vertices of T in m + 1 copies of T, and in each copy of T each edge e is embeded as an edge of the 2-factor corresponding to e. For m = 1, okay. Consider m > 1, let u be a leaf with neighbor v and T ′ = T − u. May assume uv has color k and corresponds to the 2-factor H of Gk in Fk. Let G ′ be the Cartesian product of G1, . . . , Gk−1, Gk − E(H). Consider the T ′-decomposition of G ′ provided by the induction hypothesis. Each vertex of G ′ represents v in some copy of T ′. For w ∈ V (G), let ˆ T be the copy of T ′ having v at w and let y be the vertex following w on the cycle containing w in H. Extend ˆ T by adding wy, done unless y ∈ ˆ T.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Proof

Note that F1, . . . , Fk yield a 2-factorization of G by decomposing each copy of Gi according to Fi and combining these decompositions. We prove a stronger result by induction on m. We produce a T-decomposition of G such that each vertex of G represents distinct vertices of T in m + 1 copies of T, and in each copy of T each edge e is embeded as an edge of the 2-factor corresponding to e. For m = 1, okay. Consider m > 1, let u be a leaf with neighbor v and T ′ = T − u. May assume uv has color k and corresponds to the 2-factor H of Gk in Fk. Let G ′ be the Cartesian product of G1, . . . , Gk−1, Gk − E(H). Consider the T ′-decomposition of G ′ provided by the induction hypothesis. Each vertex of G ′ represents v in some copy of T ′. For w ∈ V (G), let ˆ T be the copy of T ′ having v at w and let y be the vertex following w on the cycle containing w in H. Extend ˆ T by adding wy, done unless y ∈ ˆ T. Suppose y ∈ ˆ T, the w, y-path P in ˆ T and wy complete a cycle C in G. If C uses color i, then C collapses to a nontrivial closed trail in Gi using edges from different 2-factors in Fi, contradiction.

• Kyle F. Jao

24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

More

Theorem (J.–Kostochka–West) If every path P in T uses a color i such that Gi has no cycle consisting of edges in distinct Fi-classes all corresponding to edges of P, then G has a T-decomposition.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

More

Theorem (J.–Kostochka–West) If every path P in T uses a color i such that Gi has no cycle consisting of edges in distinct Fi-classes all corresponding to edges of P, then G has a T-decomposition. Corollary Given a list r. If T has an r-exact edge-coloring such that every path in T is 2-bounded, then G has a T-decomposition. Call such an edge-coloring 2-good.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma If T = Pm+1 and ri ≤ 2(1 +

j<i rj), then T has a 2-good r-exact

edge-coloring. Proof. 1 1

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma If T = Pm+1 and ri ≤ 2(1 +

j<i rj), then T has a 2-good r-exact

edge-coloring. Proof. 2 2 1 2 2 1 2

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma If T = Pm+1 and ri ≤ 2(1 +

j<i rj), then T has a 2-good r-exact

edge-coloring. Proof. 332332331332332331332

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma If T = Pm+1 and ri ≤ 2(1 +

j<i rj), then T has a 2-good r-exact

edge-coloring. Note that if r1 ≥ 3, then Pm+1 has NO 2-good r-exact edge-coloring.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma If T = Pm+1 and ri ≤ 2(1 +

j<i rj), then T has a 2-good r-exact

edge-coloring. Note that if r1 ≥ 3, then Pm+1 has NO 2-good r-exact edge-coloring.

• Question. Is m

k < 3 sufficient for general trees?

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma If T = Pm+1 and ri ≤ 2(1 +

j<i rj), then T has a 2-good r-exact

edge-coloring. Note that if r1 ≥ 3, then Pm+1 has NO 2-good r-exact edge-coloring.

• Question. Is m

k < 3 sufficient for general trees?

No. v k = 3, m = 8, ri = (1, 1, 6)

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma If T = Pm+1 and ri ≤ 2(1 +

j<i rj), then T has a 2-good r-exact

edge-coloring. Note that if r1 ≥ 3, then Pm+1 has NO 2-good r-exact edge-coloring.

• Question. Is m

k < 3 sufficient for general trees?

No. v k = 3, m = 8, ri = (1, 1, 6) A necessary condition is rk ≤ m − d(v).

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma If T = Pm+1 and ri ≤ 2(1 +

j<i rj), then T has a 2-good r-exact

edge-coloring. Note that if r1 ≥ 3, then Pm+1 has NO 2-good r-exact edge-coloring.

• Question. Is m

k < 3 sufficient for general trees?

No. v k = 3, m = 8, ri = (1, 1, 6) A necessary condition is ∀v rk ≤ m − d(v) + max{ℓ(v), 1} , where ℓ(v) is the number of leaf neighbors of v.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma If T = Pm+1 and ri ≤ 2(1 +

j<i rj), then T has a 2-good r-exact

edge-coloring. Note that if r1 ≥ 3, then Pm+1 has NO 2-good r-exact edge-coloring.

• Question. Is m

k < 3 sufficient for general trees?

No. v k = 3, m = 8, ri = (1, 1, 6) A necessary condition is ∀v rk ≤ m − d(v) + max{ℓ(v), 1} , where ℓ(v) is the number of leaf neighbors of v.

• Question. Is this sufficient?

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma Let T be a tree consisting of path of length at most 2 having a common

• endpoint. T has a 2-good r-exact edge-coloring if and only if

rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma Let T be a tree consisting of path of length at most 2 having a common

• endpoint. T has a 2-good r-exact edge-coloring if and only if

rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

m k < 3 and rk ≤ minv{m − d(v) + max{ℓ(v), 1}} is NOT sufficient for

general trees to have a 2-good r-exact edge-coloring.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma Let T be a tree consisting of path of length at most 2 having a common

• endpoint. T has a 2-good r-exact edge-coloring if and only if

rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

m k < 3 and rk ≤ minv{m − d(v) + max{ℓ(v), 1}} is NOT sufficient for

general trees to have a 2-good r-exact edge-coloring.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma Let T be a tree consisting of path of length at most 2 having a common

• endpoint. T has a 2-good r-exact edge-coloring if and only if

rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

m k < 3 and rk ≤ minv{m − d(v) + max{ℓ(v), 1}} is NOT sufficient for

general trees to have a 2-good r-exact edge-coloring. rk ≤ ⌈ m+1

2 ⌉ ⇒ rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma Let T be a tree consisting of path of length at most 2 having a common

• endpoint. T has a 2-good r-exact edge-coloring if and only if

rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

m k < 3 and rk ≤ minv{m − d(v) + max{ℓ(v), 1}} is NOT sufficient for

general trees to have a 2-good r-exact edge-coloring. rk ≤ ⌈ m+1

2 ⌉ ⇒ rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

Theorem (J.–Kostochka–West [2011+]) If m

k < 3 and rk ≤ ⌈ m+1 2 ⌉, then any tree has a 2-good r-exact

edge-coloring.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma Let T be a tree consisting of path of length at most 2 having a common

• endpoint. T has a 2-good r-exact edge-coloring if and only if

rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

m k < 3 and rk ≤ minv{m − d(v) + max{ℓ(v), 1}} is NOT sufficient for

general trees to have a 2-good r-exact edge-coloring. rk ≤ ⌈ m+1

2 ⌉ ⇒ rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

Theorem (J.–Kostochka–West [2011+]) If m

k < 3 and rk ≤ ⌈ m+1 2 ⌉, then any tree has a 2-good r-exact

edge-coloring.

• Question. Can we improve m

k < 3?

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

When does T have a 2-good r-exact edge-coloring?

Lemma Let T be a tree consisting of path of length at most 2 having a common

• endpoint. T has a 2-good r-exact edge-coloring if and only if

rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

m k < 3 and rk ≤ minv{m − d(v) + max{ℓ(v), 1}} is NOT sufficient for

general trees to have a 2-good r-exact edge-coloring. rk ≤ ⌈ m+1

2 ⌉ ⇒ rk ≤ minv{m − d(v) + max{ℓ(v), 1}}.

Theorem (J.–Kostochka–West [2011+]) If m

k < 3 and rk ≤ ⌈ m+1 2 ⌉, then any tree has a 2-good r-exact

edge-coloring.

• Question. Can we improve m

k < 3?

Yes, but need get rid of the paths in the main theorem.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

More results

Kouider and Lonc proved that every 2m-regular (m-reg. bip.) graph has a P4-decomposition. Theorem (J.–Kostochka–West [2011+]) If T has an r-exact edge-coloring such that every path in T is 2-bounded

• r contains a 3-bounded thread of T, then G has a T-decomposition.

(A thread in T is a path whose internal vertices have degree 2 in T.)

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

More results

Kouider and Lonc proved that every 2m-regular (m-reg. bip.) graph has a P4-decomposition. Theorem (J.–Kostochka–West [2011+]) If T has an r-exact edge-coloring such that every path in T is 2-bounded

• r contains a 3-bounded thread of T, then G has a T-decomposition.

(A thread in T is a path whose internal vertices have degree 2 in T.) Theorem (J.–Kostochka–West [2011+]) If m

k < 4 and rk ≤ ⌈ m+1 2 ⌉, then T has such an edge-coloring. Therefore,

the Cartesian product G has a T-decomposition.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring?

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54}

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 1 1

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 2 2 1 2 2 1 2 2

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 {2, 26, 26, 26}

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 {2, 26, 26, 26} 1 1

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 {2, 26, 26, 26} 2 2 1 3 3 1 4 4

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 {2, 26, 26, 26} 3 3 2 3 3 2 3 3 1 4 4 3 4 4 3 4 4 1 2 2 4 2 2 4 2 2

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 {2, 26, 26, 26} 3 3 2 3 3 2 3 3 1 4 4 3 4 4 3 4 4 1 2 2 4 2 2 4 2 2 The condition rk ≤ ⌈ m+1

2 ⌉.

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 {2, 26, 26, 26} 3 3 2 3 3 2 3 3 1 4 4 3 4 4 3 4 4 1 2 2 4 2 2 4 2 2 The condition rk ≤ ⌈ m+1

2 ⌉.

rk ≤ maxv{m − d(v) + max{ℓ(v), 1}} is nec. & suff. for singly subdivided stars (trees obtained from a vertex by attaching trees with at most two edges).

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Future Work

What is the necessary and sufficient condition for T to have a 2-good r-exact edge-coloring? The condition m

k < 4. Characterize the lists r for which Pm+1 has a

2-good r-exact edge-coloring.

• Recall. ri ≤ 2(1 +

j<i rj) ⇒ Pm+1 has a 2-good r-exact

edge-coloring. {2, 6, 18, 54} 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 1 3 3 2 3 3 2 3 3 {2, 26, 26, 26} 3 3 2 3 3 2 3 3 1 4 4 3 4 4 3 4 4 1 2 2 4 2 2 4 2 2 The condition rk ≤ ⌈ m+1

2 ⌉.

rk ≤ maxv{m − d(v) + max{ℓ(v), 1}} is nec. & suff. for singly subdivided stars (trees obtained from a vertex by attaching trees with at most two edges). Find a nec. & suff. condition for trees obtained from a vertex by attaching trees with at most q edges?

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Thank you!

Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing