Gallai-Colorings of Triples and 2-Factors of

Lynn Chua,András Gyárfás,Chetak Hossain

doi:10.1155/2013/929565

Abstract

A coloring of the edges of the -uniform complete hypergraph is a -coloring if there is no rainbow simplex; that is, every set of vertices contains two edges of the same color. The notion extends -colorings which are often called Gallai-colorings and originates from a seminal paper of Gallai. One well-known property of -colorings is that at least one color class has a spanning tree. J. Lehel and the senior author observed that this property does not hold for -colorings and proposed to study , the size of the largest monochromatic component which can be found in every -coloring of , the complete -uniform hypergraph. The previous remark says that and in this note, we address the case . We prove that and this determines for . We also prove that by excluding certain 2-factors from the middle layer of the Boolean lattice on seven elements.

Highlights

Studying edge colorings of complete graphs without rainbow triangles originates from a famous paper of Gallai [1]; we will refer to such colorings as Gallai-colorings or G2-colorings
A well-known property of G2-colorings is that the edges of some color class span a connected subgraph containing all vertices
A natural extension of the concept is the Gr-coloring, of Knr, the complete r-uniform hypergraph on requirement is that no Krr+1 n is vertices

Summary

Introduction

Studying edge colorings of complete graphs without rainbow triangles (no triangles colored with distinct colors) originates from a famous paper of Gallai [1]; we will refer to such colorings as Gallai-colorings or G2-colorings. If the lower bound of Theorem 1 is sharp for some odd n ≥ 5 there is a (2, (n − 3)/2)-factor F in Bn4,3, such that the union of the triples and quadruples in every component of F has at most (n + 3)/2 elements.

Results

Conclusion