A Note on Path Factors of (3,4)-Biregular Bipartite Graphs

A proper edge coloring of a graph G with colors 1, 2, 3, . . . is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graph is (3, 4)-biregular if all vertices in one part have degree 3 and all vertices in the other part have degree 4. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph G has a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}, then G has an interval coloring. It was also conjectured that every simple (3, 4)-biregular bipartite graph has such a subgraph. We provide some evidence for this conjecture by proving that a simple (3, 4)-biregular bipartite graph has a spanning subgraph whose components are nontrivial paths with endpoints at 3-valent vertices and lengths not exceeding 22.