Properly colored subgraphs and rainbow subgraphs in edge‐colorings with local constraints

Abstract

ABSTRACT:We consider a canonical Ramsey type problem. An edge-coloring of a graph is called m-good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m-good edge-coloring of K n yields a properly edge-colored copy of G, and let g(m, G) denote the smallest n such that every m-good edgecoloring of K n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G ϭ K t , we have c 1 mt 2 /ln t Յ f(m, K t ) Յ c 2 mt 2 , and cЈ 1 mt 3 /ln t Յ g(m, K t ) Յ cЈ 2 mt 3 /ln t, where c 1 , c 2 , cЈ 1 , cЈ 2 are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) ϭ n for all graphs G with n vertices and maximum degree at most d.

Properly colored and rainbow copies of graphs with few cherries

“…This time we show that t vj 6 (v) is at most e(G) 2 (n − − 1)k for every j ∈ [4]. Without loss of generality, v = v 1 .…”

“…Another generalization of the conjecture of Bollobás and Erdős to a general graph G takes into account the maximum degree. Alon, Jiang, Miller and Pritikin [4] showed that if G is an n-vertex graph with maximum degree ∆ and k = O √ n ∆ 27/2 , then any locally k-bounded coloring c of E(K n ) is G-proper. Their result was greatly improved by Böttcher, Kohayakawa and Procacci [6] who showed that k can be of order n/∆ 2 .…”

“…. , B 10 split the whole set B follows because if u i ∈ L for some i ∈ [4], then u 1 ∈ L, and also if u 4 ∈ L, then u 3 ∈ L. It holds that…”

On degree anti-Ramsey numbers

“…One such upper bound was proved by Alon, Jiang, Miller and Pritikin in [3]. It was proved there that for every graph H,…”

Rainbow decompositions

“…Jamison, Jiang and Ling [7], and Chen, Schelp and Wei [4] considered Ramsey type variants where an arbitrary number of colors can be used. Alon et al [1] studied the function f (H) which is the minimum integer n such that any proper edge coloring of K n has a rainbow copy of H. Keevash et al [8] considered the rainbow Turán number ex * (n, H) which is the largest integer m such that there exists a properly edge-colored graph with n vertices and m edges and which has no rainbow copy of H. Yuster [12] gave necessary and sufficient conditions for the existence of rainbow H-factors.…”