Strict neighbor-distinguishing index of K4-minor-free graphs

A proper edge-coloring of a graph G is strict neighbor-distinguishing if for any two adjacent vertices u and v, the set of colors used on the edges incident with u and the set of colors used on the edges incident with v are not included in each other. The strict neighbor-distinguishing index χsnd′(G) of G is the minimum number of colors in a strict neighbor-distinguishing edge-coloring of G. A graph is formal if its minimum degree is at least 2. Let Hn denote the graph obtained from the complete bipartite graph K2,n by inserting a 2-vertex into one edge. In this paper, we prove that if G is a formal K4-minor-free graph, then χsnd′(G)≤2Δ+1, and moreover χsnd′(G)=2Δ+1 if and only if G is HΔ. This shows partially a conjecture, which says that every formal graph G, different from HΔ, has χsnd′(G)≤2Δ.