Rainbow Induced Subgraphs in Proper Vertex Colorings

Abstract

Given a graph H we define ρ(H) to be the minimum order of a graph G such that every proper vertex coloring of G contains a rainbow induced subgraph isomorphic to H. We give upper and lower bounds for ρ(H), compute the exact value for some classes of graphs, and consider an interesting combinatorial problem connected with computation of ρ(H) for paths. A part of this research has been guided by a computer search and, accordingly, some computational results are presented. A special motivation comes from research in on-line coloring.