On improper interval colorings of complete multipartite graphs

Abstract

An improper edge-coloring of a graph [Formula: see text] is a mapping [Formula: see text]. An improper edge-coloring of [Formula: see text] is called an improper interval coloring if the colors (excluding repetitions) of the edges incident to each vertex of [Formula: see text] form an integral interval. An improper coloring of [Formula: see text] is called [Formula: see text]-improper coloring, if there are at most [Formula: see text] adjacent edges with the same color. In this note we show that complete multipartite graphs have a [Formula: see text]-improper interval coloring; this proves a conjecture of Casselgren and Petrosyan.