Forbidden rainbow subgraphs that force large monochromatic or multicolored [formula omitted]-connected subgraphs

Abstract

Let n,k,m be positive integers with n≫m≫k, and let G (respectively, H) be the set of connected (respectively, disconnected) graphs G such that there is a k-connected monochromatic subgraph of order at least n−f(G,k,m) in every rainbow G-free m-coloring of Kn, where f(G,k,m) does not depend on n. The set G was determined by Fujita and Magnant in a previous paper. In this paper, we give a complete characterization of H, and we show that G∪H consists of precisely P6, P3∪P4, K2∪P5, K2∪2P3, 2K2∪K3, 2K2∪P4+, 3K2∪K1,3 and their subgraphs.We also consider a parallel problem for complete bipartite graphs. Let s,t,k,m be positive integers with mins,t≫m≫k and m≥|E(H)|, and let B be the set of bipartite graphs H such that there is a k-connected monochromatic subgraph of order at least s+t−f(H,k,m) in any rainbow H-free m-coloring of Ks,t, where f(H,k,m) does not depend on s or t. We prove that the set B consists of precisely 2P3, 2K2∪K1,3 and their subgraphs.Finally, we consider large k-connected multicolored subgraphs instead of monochromatic subgraphs. We show that for 1≤k≤3 and sufficiently large n, every Gallai-3-coloring of Kn contains a k-connected subgraph of order at least n−k−12 using at most two colors. For any positive integer t, we also show that the above statement is false for k=4t.