On interval and cyclic interval edge colorings of (3,5)-biregular graphs

A proper edge coloring f of a graph G with colors 1,2,3,…,t is called an interval coloring if the colors on the edges incident to every vertex of G form an interval of integers. The coloring f is cyclic interval if for every vertex v of G, the colors on the edges incident to v either form an interval or the set {1,…,t}∖{f(e):e is incident to v} is an interval. A bipartite graph G 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 every (3,5)-biregular graph has a cyclic interval coloring and we give several sufficient conditions for a (3,5)-biregular graph to admit an interval coloring.