Interval coloring of (3, 4)-biregular bigraphs having two -biregular bipartite subgraphs

An interval coloring of a graph is a proper edge coloring such that the set of colors used at every vertex is an interval of integers. An ( α , β )-biregular bigraph is a bipartite graph in which each vertex in one part has degree α and each vertex in the other part has degree β . It is unknown whether all ( 3 , 4 ) -biregular bigraphs have interval colorings. In this work we prove that if a ( 3 , 4 ) -biregular bigraph G = ( X , Y ; E ) has two (2, 3)-biregular bipartite subgraphs G 1 = ( Y , X 1 ; E 1 ) , G 2 = ( Y , X 2 ; E 2 ) such that X 1 ∪ X 2 = X , E 1 ∪ E 2 = E , X 1 ∩ X 2 = 0̸ , and E 1 ∩ E 2 = 0̸ , then G has an interval 6-coloring.