On Decomposition of Cartesian Products ofRegular Graphs into Isomorphic Trees

Kyle F. Jao

Department of MathematicsUniversity 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 ifthe 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–Haggkvist [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

The Problem

Let T be a fixed tree with m edges. A graph G has a T-decomposition ifthe 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–Haggkvist [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

The Problem

Let T be a fixed tree with m edges. A graph G has a T-decomposition ifthe 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–Haggkvist [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–Haggkvist conjecture for the casegirth(G ) > diam(T ).

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

Results

Snevily gave a short proof for Graham–Haggkvist conjecture for the casegirth(G ) > diam(T ).

Theorem (Snevily [1991])

Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with1-fact.). If G has no cycle with length at most diam(T ) consisting ofedges 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–Haggkvist conjecture for the casegirth(G ) > diam(T ).

Theorem (Snevily [1991])

Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with1-fact.). If G has no cycle with length at most diam(T ) consisting ofedges 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–Haggkvist conjecture for the casegirth(G ) > diam(T ).

Theorem (Snevily [1991])

Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with1-fact.). If G has no cycle with length at most diam(T ) consisting ofedges in distinct F-classes, then G has a T-decomposition.

Let G be the Cartesian product of G1, . . . ,Gk , where Gi is a 2ri -regulargraph 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–Haggkvist conjecture for the casegirth(G ) > diam(T ).

Theorem (Snevily [1991])

Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with1-fact.). If G has no cycle with length at most diam(T ) consisting ofedges in distinct F-classes, then G has a T-decomposition.

Let G be the Cartesian product of G1, . . . ,Gk , where Gi is a 2ri -regulargraph 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 anr -exact edge-coloring of T and establish a one-to-one correspondencebetween 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–Haggkvist conjecture for the casegirth(G ) > diam(T ).

Theorem (Snevily [1991])

Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with1-fact.). If G has no cycle with length at most diam(T ) consisting ofedges in distinct F-classes, then G has a T-decomposition.

Let G be the Cartesian product of G1, . . . ,Gk , where Gi is a 2ri -regulargraph 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 anr -exact edge-coloring of T and establish a one-to-one correspondencebetween 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 ofedges in distinct Fi -classes all corresponding to edges of P, then G has aT-decomposition.

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

Results

Snevily gave a short proof for Graham–Haggkvist conjecture for the casegirth(G ) > diam(T ).

Theorem (Snevily [1991])

Let G be a 2m-regular graph with a 2-factorization F (m-reg. bip. with1-fact.). If G has no cycle with length at most diam(T ) consisting ofedges in distinct F-classes, then G has a T-decomposition.

Let G be the Cartesian product of G1, . . . ,Gk , where Gi is a 2ri -regulargraph 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 anr -exact edge-coloring of T and establish a one-to-one correspondencebetween 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 ofedges in distinct Fi -classes all corresponding to edges of P, then G has aT-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 eachcopy of Gi according to Fi and combining these decompositions.

We prove a stronger result by induction on m. We produce aT -decomposition of G such that each vertex of G represents distinctvertices of T in m + 1 copies of T , and in each copy of T each edge e isembeded 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 andT ′ = T − u. May assume uv has color k and corresponds to the 2-factorH of Gk in Fk .

Let G ′ be the Cartesian product of G1, . . . ,Gk−1, Gk − E (H). Considerthe T ′-decomposition of G ′ provided by the induction hypothesis. Eachvertex of G ′ represents v in some copy of T ′.

For w ∈ V (G ), let T be the copy of T ′ having v at w and let y be thevertex following w on the cycle containing w in H. Extend T by addingwy , done unless y ∈ T .

Suppose y ∈ T , the w , y -path P in T and wy complete a cycle C in G .If C uses color i , then C collapses to a nontrivial closed trail in Gi usingedges from different 2-factors in Fi , contradiction. �

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 eachcopy of Gi according to Fi and combining these decompositions.

We prove a stronger result by induction on m. We produce aT -decomposition of G such that each vertex of G represents distinctvertices of T in m + 1 copies of T , and in each copy of T each edge e isembeded 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 andT ′ = T − u. May assume uv has color k and corresponds to the 2-factorH of Gk in Fk .

Let G ′ be the Cartesian product of G1, . . . ,Gk−1, Gk − E (H). Considerthe T ′-decomposition of G ′ provided by the induction hypothesis. Eachvertex of G ′ represents v in some copy of T ′.

For w ∈ V (G ), let T be the copy of T ′ having v at w and let y be thevertex following w on the cycle containing w in H. Extend T by addingwy , done unless y ∈ T .

Suppose y ∈ T , the w , y -path P in T and wy complete a cycle C in G .If C uses color i , then C collapses to a nontrivial closed trail in Gi usingedges from different 2-factors in Fi , contradiction. �

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 eachcopy of Gi according to Fi and combining these decompositions.

We prove a stronger result by induction on m. We produce aT -decomposition of G such that each vertex of G represents distinctvertices of T in m + 1 copies of T , and in each copy of T each edge e isembeded 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 andT ′ = T − u. May assume uv has color k and corresponds to the 2-factorH of Gk in Fk .

Let G ′ be the Cartesian product of G1, . . . ,Gk−1, Gk − E (H). Considerthe T ′-decomposition of G ′ provided by the induction hypothesis. Eachvertex of G ′ represents v in some copy of T ′.

For w ∈ V (G ), let T be the copy of T ′ having v at w and let y be thevertex following w on the cycle containing w in H. Extend T by addingwy , done unless y ∈ T .

Suppose y ∈ T , the w , y -path P in T and wy complete a cycle C in G .If C uses color i , then C collapses to a nontrivial closed trail in Gi usingedges from different 2-factors in Fi , contradiction. �

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 eachcopy of Gi according to Fi and combining these decompositions.

We prove a stronger result by induction on m. We produce aT -decomposition of G such that each vertex of G represents distinctvertices of T in m + 1 copies of T , and in each copy of T each edge e isembeded 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 andT ′ = T − u. May assume uv has color k and corresponds to the 2-factorH of Gk in Fk .

Let G ′ be the Cartesian product of G1, . . . ,Gk−1, Gk − E (H). Considerthe T ′-decomposition of G ′ provided by the induction hypothesis. Eachvertex of G ′ represents v in some copy of T ′.

For w ∈ V (G ), let T be the copy of T ′ having v at w and let y be thevertex following w on the cycle containing w in H. Extend T by addingwy , done unless y ∈ T .

Suppose y ∈ T , the w , y -path P in T and wy complete a cycle C in G .If C uses color i , then C collapses to a nontrivial closed trail in Gi usingedges from different 2-factors in Fi , contradiction. �

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 eachcopy of Gi according to Fi and combining these decompositions.

We prove a stronger result by induction on m. We produce aT -decomposition of G such that each vertex of G represents distinctvertices of T in m + 1 copies of T , and in each copy of T each edge e isembeded 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 andT ′ = T − u. May assume uv has color k and corresponds to the 2-factorH of Gk in Fk .

Let G ′ be the Cartesian product of G1, . . . ,Gk−1, Gk − E (H). Considerthe T ′-decomposition of G ′ provided by the induction hypothesis. Eachvertex of G ′ represents v in some copy of T ′.

For w ∈ V (G ), let T be the copy of T ′ having v at w and let y be thevertex following w on the cycle containing w in H. Extend T by addingwy , done unless y ∈ T .

Suppose y ∈ T , the w , y -path P in T and wy complete a cycle C in G .If C uses color i , then C collapses to a nontrivial closed trail in Gi usingedges from different 2-factors in Fi , contradiction. �

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 eachcopy of Gi according to Fi and combining these decompositions.

We prove a stronger result by induction on m. We produce aT -decomposition of G such that each vertex of G represents distinctvertices of T in m + 1 copies of T , and in each copy of T each edge e isembeded 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 andT ′ = T − u. May assume uv has color k and corresponds to the 2-factorH of Gk in Fk .

Let G ′ be the Cartesian product of G1, . . . ,Gk−1, Gk − E (H). Considerthe T ′-decomposition of G ′ provided by the induction hypothesis. Eachvertex of G ′ represents v in some copy of T ′.

For w ∈ V (G ), let T be the copy of T ′ having v at w and let y be thevertex following w on the cycle containing w in H. Extend T by addingwy , done unless y ∈ T .

Suppose y ∈ T , the w , y -path P in T and wy complete a cycle C in G .If C uses color i , then C collapses to a nontrivial closed trail in Gi usingedges 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 ofedges in distinct Fi -classes all corresponding to edges of P, then G has aT-decomposition.

Corollary

Given a list r . If T has an r-exact edge-coloring such that every path inT 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

More

Theorem (J.–Kostochka–West)

If every path P in T uses a color i such that Gi has no cycle consisting ofedges in distinct Fi -classes all corresponding to edges of P, then G has aT-decomposition.

Corollary

Given a list r . If T has an r-exact edge-coloring such that every path inT 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?

Lemma

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

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

Question. Is mk < 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{`(v), 1} , where `(v) isthe 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

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

Proof. 1 1

Note that if r1 ≥ 3, then Pm+1

has NO 2-good r -exact edge-coloring.

Question. Is mk < 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{`(v), 1} , where `(v) isthe 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

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

Proof. 2 2 1 2 2 1 2

Note that if r1 ≥ 3, then Pm+1

has NO 2-good r -exact edge-coloring.

Question. Is mk < 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{`(v), 1} , where `(v) isthe 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

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

Proof. 332332331332332331332 �

Note that if r1 ≥ 3, then Pm+1

has NO 2-good r -exact edge-coloring.

Question. Is mk < 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{`(v), 1} , where `(v) isthe 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

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

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

Question. Is mk < 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{`(v), 1} , where `(v) isthe 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

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

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

Question. Is mk < 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{`(v), 1} , where `(v) isthe 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

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

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

Question. Is mk < 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{`(v), 1} , where `(v) isthe 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

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

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

Question. Is mk < 3 sufficient for general trees? No.

v

k = 3, m = 8, ri = (1, 1, 6)

A necessary condition is rk ≤ m − d(v).

A necessary condition is ∀vrk ≤ m − d(v) + max{`(v), 1} , where `(v) is the number of leafneighbors 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

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

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

Question. Is mk < 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{`(v), 1} , where `(v) isthe 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

If T = Pm+1 and ri ≤ 2(1 +∑

j<i rj), then T has a 2-good r-exactedge-coloring.

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

Question. Is mk < 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{`(v), 1} , where `(v) isthe 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 commonendpoint. T has a 2-good r-exact edge-coloring if and only ifrk ≤ minv{m − d(v) + max{`(v), 1}}.

mk < 3 and rk ≤ minv{m − d(v) + max{`(v), 1}} is NOT sufficient forgeneral trees to have a 2-good r -exact edge-coloring.rk ≤ dm+1

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

Theorem (J.–Kostochka–West [2011+])

If mk < 3 and rk ≤ dm+1

2 e, then any tree has a 2-good r-exactedge-coloring.

Question. Can we improve mk < 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

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 commonendpoint. T has a 2-good r-exact edge-coloring if and only ifrk ≤ minv{m − d(v) + max{`(v), 1}}.

mk < 3 and rk ≤ minv{m − d(v) + max{`(v), 1}} is NOT sufficient forgeneral trees to have a 2-good r -exact edge-coloring.

mk < 3 and rk ≤ minv{m − d(v) + max{`(v), 1}} is NOT sufficient forgeneral trees to have a 2-good r -exact edge-coloring.rk ≤ dm+1

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

Theorem (J.–Kostochka–West [2011+])

If mk < 3 and rk ≤ dm+1

2 e, then any tree has a 2-good r-exactedge-coloring.

Question. Can we improve mk < 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

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 commonendpoint. T has a 2-good r-exact edge-coloring if and only ifrk ≤ minv{m − d(v) + max{`(v), 1}}.

mk < 3 and rk ≤ minv{m − d(v) + max{`(v), 1}} is NOT sufficient forgeneral trees to have a 2-good r -exact edge-coloring.

rk ≤ dm+12 e ⇒ rk ≤ minv{m − d(v) + max{`(v), 1}}.

Theorem (J.–Kostochka–West [2011+])

If mk < 3 and rk ≤ dm+1

2 e, then any tree has a 2-good r-exactedge-coloring.

Question. Can we improve mk < 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

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 commonendpoint. T has a 2-good r-exact edge-coloring if and only ifrk ≤ minv{m − d(v) + max{`(v), 1}}.

mk < 3 and rk ≤ minv{m − d(v) + max{`(v), 1}} is NOT sufficient forgeneral trees to have a 2-good r -exact edge-coloring.rk ≤ dm+1

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

Theorem (J.–Kostochka–West [2011+])

If mk < 3 and rk ≤ dm+1

2 e, then any tree has a 2-good r-exactedge-coloring.

Question. Can we improve mk < 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

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 commonendpoint. T has a 2-good r-exact edge-coloring if and only ifrk ≤ minv{m − d(v) + max{`(v), 1}}.

mk < 3 and rk ≤ minv{m − d(v) + max{`(v), 1}} is NOT sufficient forgeneral trees to have a 2-good r -exact edge-coloring.rk ≤ dm+1

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

Theorem (J.–Kostochka–West [2011+])

If mk < 3 and rk ≤ dm+1

2 e, then any tree has a 2-good r-exactedge-coloring.

Question. Can we improve mk < 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

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 commonendpoint. T has a 2-good r-exact edge-coloring if and only ifrk ≤ minv{m − d(v) + max{`(v), 1}}.

mk < 3 and rk ≤ minv{m − d(v) + max{`(v), 1}} is NOT sufficient forgeneral trees to have a 2-good r -exact edge-coloring.rk ≤ dm+1

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

Theorem (J.–Kostochka–West [2011+])

If mk < 3 and rk ≤ dm+1

2 e, then any tree has a 2-good r-exactedge-coloring.

Question. Can we improve mk < 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

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 commonendpoint. T has a 2-good r-exact edge-coloring if and only ifrk ≤ minv{m − d(v) + max{`(v), 1}}.

mk < 3 and rk ≤ minv{m − d(v) + max{`(v), 1}} is NOT sufficient forgeneral trees to have a 2-good r -exact edge-coloring.rk ≤ dm+1

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

Theorem (J.–Kostochka–West [2011+])

If mk < 3 and rk ≤ dm+1

2 e, then any tree has a 2-good r-exactedge-coloring.

Question. Can we improve mk < 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 hasa P4-decomposition.

Theorem (J.–Kostochka–West [2011+])

If T has an r-exact edge-coloring such that every path in T is 2-boundedor 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 mk < 4 and rk ≤ dm+1

2 e, 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

More results

Kouider and Lonc proved that every 2m-regular (m-reg. bip.) graph hasa P4-decomposition.

Theorem (J.–Kostochka–West [2011+])

If T has an r-exact edge-coloring such that every path in T is 2-boundedor 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 mk < 4 and rk ≤ dm+1

2 e, 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-goodr -exact edge-coloring?

The condition mk < 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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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 -exactedge-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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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

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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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

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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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

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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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

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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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

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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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

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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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 ≤ dm+12 e.

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

Find a nec. & suff. condition for trees obtained from a vertex byattaching trees with at most q 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-goodr -exact edge-coloring?

The condition mk < 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 ≤ dm+12 e.

rk ≤ maxv{m − d(v) + max{`(v), 1}} is nec. & suff. for singlysubdivided stars (trees obtained from a vertex by attaching treeswith at most two edges).Find a nec. & suff. condition for trees obtained from a vertex byattaching 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