On one-sided interval edge colorings of biregular bipartite graphs

Abstract

A proper edge t-coloring of a graph G is a coloring of edges of G with colors 1,2,…,t such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex x is called a spectrum of x. Any nonempty subset of consecutive integers is called an interval. A proper edge t-coloring of a graph G is interval in the vertex x if the spectrum of x is an interval. A proper edge t-coloring φ of a graph G is interval on a subset R0 of vertices of G, if for any x∈R0, φ is interval in x. A subset R of vertices of G has an i-property if there is a proper edge t-coloring of G which is interval on R. If G is a graph, and a subset R of its vertices has an i-property, then the minimum value of t for which there is a proper edge t-coloring of G interval on R is denoted by wR(G). We estimate the value of this parameter for biregular bipartite graphs in the case when R is one of the sides of a bipartition of the graph