Split and balanced colorings of complete graphs Original Research Article

An (r, n)-split coloring of a complete graph is an edge coloring with r colors under which the vertex set is partitionable into r parts so that for each i, part i does not contain Kn in color i. This generalizes the notion of split graphs which correspond to (2, 2)-split colorings. The smallest N for which the complete graph KN has a coloring which is not (r, n)-split is denoted by ƒr(n). Balanced (r,n)-colorings are defined as edge r-colorings of KN such that every subset of [N/r] vertices contains a monochromatic Kn in all colors. Then gr(n) is defined as the smallest N such that KN has a balanced (r, n)-coloring. The definitions imply that fr(n) ⩽ gr(n). The paper gives estimates and exact values of these functions for various choices of parameters.