Note on injective edge-coloring of graphs

Abstract

An injective edge-coloring of graph G is an edge-coloring φ such that φ(e1)≠φ(e3) for any three consecutive edges e1, e2 and e3 of a path or a 3-cycle. Note that such an edge-coloring is not necessarily proper. The minimum number of colors required for an injective edge-coloring is called the injective chromatic index of G, denoted by χi′(G). For every integer k≥2, we show that every k-degenerate graph G with maximum degree Δ satisfies χi′(G)≤(4k−3)Δ−2k2−k+3. We also prove that every graph G with Δ=4, it is injective 9-edge-colorable when its maximum average degree mad(G)<145, injective 10-edge-colorable when mad(G)<3, injective 11-edge-colorable when mad(G)<196, and injective 12-edge-colorable when mad(G)<3611.