Color-blind index in graphs of very low degree

Abstract

Let c:E(G)→[k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v)=(a1,…,ak), where ai is the number of edges incident to v with color i. Reorder c̄(v) for every v in G in nonincreasing order to obtain c∗(v), the color-blind partition of v. When c∗ induces a proper vertex coloring, that is, c∗(u)≠c∗(v) for every edge uv in G, we say that c is color-blind distinguishing. The minimum k for which there exists a color-blind distinguishing edge coloring c:E(G)→[k] is the color-blind index of G, denoted dal(G). We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if dal(G)≤2 is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when dal(G) is finite for a class of 3-regular graphs.