Injective Chromatic Index of $$K_4$$ -Minor Free Graphs

Abstract

An edge-coloring of a graph G is injective if for any two distinct edges \(e_1\) and \(e_2\), the colors of \(e_1\) and \(e_2\) are distinct if they are at distance 2 in G or in a common triangle. The injective chromatic index of G, \(\chi ^\prime _{inj}(G)\), is the minimum number of colors needed for an injective edge-coloring of G. In this note, we show that every \(K_4\)-minor free graph G with maximum degree \(\Delta (G)\ge 3\) satisfies \(\chi ^\prime _{inj}(G)\le 2\Delta (G)+1\).