Interval coloring of (3, 4)-biregular bigraphs having two (2,3)(2,3)-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,βα,β)-biregular bigraph is a bipartite graph in which each vertex in one part has degree αα and each vertex in the other part has degree ββ. It is unknown whether all (3,4)(3,4)-biregular bigraphs have interval colorings. In this work we prove that if a (3,4)(3,4)-biregular bigraph G=(X,Y;E)G=(X,Y;E) has two (2, 3)-biregular bipartite subgraphs G1=(Y,X1;E1)G1=(Y,X1;E1), G2=(Y,X2;E2)G2=(Y,X2;E2) such that X1∪X2=XX1∪X2=X, E1∪E2=EE1∪E2=E, X1∩X2=0̸X1∩X2=0̸, and E1∩E2=0̸E1∩E2=0̸, then GG has an interval 6-coloring.