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.