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

A proper edge coloring of a graph $G$ with colors $1,2,3,\dots$ 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 $22$.