On Interval Edge Colorings of Biregular Bipartite Graphs With Small Vertex Degrees

A proper edge coloring of a graph with colors 1, 2, 3, ... is called an interval coloring if the colors on the edges incident to each vertex form an interval of integers. A bipartite graph is (a,b)-biregular if every vertex in one part has degree a and every vertex in the other part has degree b. It has been conjectured that all such graphs have interval colorings. We prove that all (3, 6)-biregular graphs have interval colorings and that all (3, 9)-biregular graphs having a cubic subgraph covering all vertices of degree 9 admit interval colorings. Moreover, we prove that slightly weaker versions of the conjecture hold for (3, 5)-biregular, (4, 6)-biregular, and (4, 8)-biregular graphs. All our proofs are constructive and yield polynomial time algorithms for constructing the required colorings. (C) 2014 Wiley Periodicals, Inc.