On one-sided interval edge colorings of biregular bipartite graphs

A proper edge t-coloring of a graphG 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. We consider undirected, finite graphs without loops and multiple edges. V (G) and E(G) denote the sets of vertices and edges of a graph G, respectively. For any vertex x ∈ V (G), we denote by NG(x) the set of vertices of a graph G adjacent to x. The degree of a vertex x of a graph G is denoted by dG(x), the maximum degree of a vertex of G by ∆(G). For a graph G and an arbitrary subset V0 ⊆ V (G), we denote by G[V0] the subgraph of G induced by the subset V0 of its vertices. 2010 MSC: 05C15, 05C50, 05C85.