Color-blind index in graphs of very low degree

Jennifer Diemunsch, Nathan Graber, Lucas Kramer, Victor Larsen, Lauren M. Nelsen, Luke L. Nelsen, Devon Sigler, Derrick Stolee, Charlie Suer
2017 Discrete Applied Mathematics  
Let c:E(G)→ [k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̅(v)=(a_1,...,a_k), where a_i is the number of edges incident to v with color i. Reorder c̅(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
more » ... :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.
doi:10.1016/j.dam.2017.03.006 fatcat:wk6gbsunsnh5tfanmndagzmfqy