Star edge-coloring of graphs with maximum degree four

The star chromatic index χst′(G) of a graph G is the smallest integer k for which G has a proper k-edge-coloring without bichromatic paths or cycles of length four. In this paper, we prove that (1) if G is a graph with Δ=4, then χst′(G)≤14; and (2) if G is a bipartite graph with Δ=4, then χst′(G)≤13.