Let $c:E(G)to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $bar{c}(v)=(a_1,ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $bar{c}(v)$ 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, $c^*(u) eq c^*(v)$ 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 $c:E(G)to [k]$ is the color-blind index of $G$, denoted $operatorname{dal}(G)$. We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if $operatorname{dal}(G) leq 2$ is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when $operatorname{dal}(G)$ is finite for a class of 3-regular graphs.
Download