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 H1 ⊆ H such that | V (H1) | ≥ c | V (H) | where c depends only on F, r, and α (H).
- Gallai colorings
- Monochromatic connected components
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics