Edge colorings avoiding patterns

Abstract

We say that a pattern is a graph together with an edge coloring, and a pattern P=(H,c) occurs in some edge coloring c′ of G if c′, restricted to some subgraph of G isomorphic to H, is equal to c up to renaming the colors. Inspired by Matoušek’s visibility blocking problem, we study edge colorings that avoid certain patterns.We show that for every pattern P, such that the number of edges in P is at least the number of vertices in P plus the number of colors minus 2, there is a constant C such that every graph with maximum degree Δ admits an edge coloring with CΔ colors avoiding P; the same also holds for infinite sets of such patterns, provided that the number of patterns in the set grows at most exponentially.