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]) K 2 m +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]) K 2 m +1 has a T-decomposition. Conjecture (Graham–H¨ aggkvist [1984]) Every 2 m-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 2 m-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 2 m-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 2 m-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 G 1 , . . . , G k , where G i is a 2 r i -regular graph with a 2-factorization F i (or r i -reg. bip. with 1-fact.). Say r = ( r 1 , . . . , r k ) with r 1 ≤ . . . ≤ r k 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 2 m-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 G 1 , . . . , G k , where G i is a 2 r i -regular graph with a 2-factorization F i (or r i -reg. bip. with 1-fact.). Say r = ( r 1 , . . . , r k ) with r 1 ≤ . . . ≤ r k and sum m . An edge-coloring of T is r -exact if exactly r i 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 F i 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 2 m-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 G 1 , . . . , G k , where G i is a 2 r i -regular graph with a 2-factorization F i (or r i -reg. bip. with 1-fact.). Say r = ( r 1 , . . . , r k ) with r 1 ≤ . . . ≤ r k and sum m . An edge-coloring of T is r -exact if exactly r i 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 F i for each i . Theorem (J.–Kostochka–West [2011+]) If every path P in T uses a color i such that G i has no cycle consisting of edges in distinct F i -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 2 m-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 G 1 , . . . , G k , where G i is a 2 r i -regular graph with a 2-factorization F i (or r i -reg. bip. with 1-fact.). Say r = ( r 1 , . . . , r k ) with r 1 ≤ . . . ≤ r k and sum m . An edge-coloring of T is r -exact if exactly r i 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 F i for each i . Theorem (J.–Kostochka–West [2011+]) If every path P in T uses a color i such that G i has no cycle consisting of edges in distinct F i -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 F 1 , . . . , F k yield a 2-factorization of G by decomposing each copy of G i according to F i and combining these decompositions. Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Proof Note that F 1 , . . . , F k yield a 2-factorization of G by decomposing each copy of G i according to F i 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 F 1 , . . . , F k yield a 2-factorization of G by decomposing each copy of G i according to F i 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 G k in F k . Kyle F. Jao 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Proof Note that F 1 , . . . , F k yield a 2-factorization of G by decomposing each copy of G i according to F i 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 G k in F k . Let G ′ be the Cartesian product of G 1 , . . . , G k − 1 , G k − 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 F 1 , . . . , F k yield a 2-factorization of G by decomposing each copy of G i according to F i 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 G k in F k . Let G ′ be the Cartesian product of G 1 , . . . , G k − 1 , G k − E ( H ). Consider the T ′ -decomposition of G ′ provided by the induction hypothesis. Each vertex of G ′ represents v in some copy of T ′ . T be the copy of T ′ having v at w and let y be the For w ∈ V ( G ), let ˆ 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