Gallai colorings of non-complete graphs

Abstract

Gallai-colorings of complete graphs–edge colorings such that no triangle is colored with three distinct colors–occur in various contexts such as the theory of partially ordered sets (in Gallai’s original paper), information theory and the theory of perfect graphs. We extend here Gallai-colorings to non-complete graphs and study the analogue of a basic result–any Gallai-colored complete graph has a monochromatic spanning tree–in this more general setting. We show that edge colorings of a graph H without multicolored triangles contain monochromatic connected subgraphs with at least ( α ( H ) 2 + α ( H ) − 1 ) − 1 | V ( H ) | vertices, where α ( H ) is the independence number of H . In general, we show that if the edges of an r -uniform hypergraph H are colored so that there is no multicolored copy of a fixed F then there is a monochromatic connected subhypergraph H 1 ⊆ H such that | V ( H 1 ) | ≥ c | V ( H ) | where c depends only on F , r , and α ( H ) .