Note on injective edge-coloring of graphs

An injective edge-coloring of graph G is an edge-coloring phi such that phi(e(1)) not equal (e(3)) for any three consecutive edges e(1), e(2) and e(3) 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 chi(i)'(G). For every integer k >= 2, we show that every k-degenerate graph G with maximum degree Delta satisfies chi(i)'(G) <= (4k- 3)Delta - 2k(2) - k+3. We also prove that every graph G with Delta = 4, it is injective 9-edge-colorable when its maximum average degree mad(G) < 14/5, injective 10-edge-colorable when mad(G) < 3, injective 11-edge-colorable when mad(G) < 19/6, and injective 12-edge-colorable when mad(G) < 36/11. (C) 2021 Elsevier B.V. All rights reserved.