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Let $c:E(G)\to [k]$ be an edge-coloring of a graph $G$, not necessarilyproper. For each vertex $v$, let $\bar{c}(v)=(a_1,\ldots,a_k)$, where $a_i$ isthe number of edges incident to $v$ with color $i$. Reorder $\bar{c}(v)$ forevery $v$ in $G$ in nonincreasing order to obtain $c^*(v)$, the color-blindpartition of $v$. When $c^*$ induces a proper vertex coloring, that is,$c^*(u)\neq c^*(v)$ for every edge $uv$ in $G$, we say that $c$ is color-blinddistinguishing. The minimum $k$ for which there exists a color-blinddistinguishing edge coloring $c:E(G)\to [k]$ is the color-blind index of $G$,denoted $\operatorname{dal}(G)$. We demonstrate that determining thecolor-blind index is more subtle than previously thought. In particular,determining if $\operatorname{dal}(G) \leq 2$ is NP-complete. We also connectthe color-blind index of a regular bipartite graph to 2-colorable regularhypergraphs and characterize when $\operatorname{dal}(G)$ is finite for a classof 3-regular graphs.

Color-blind index in graphs of very low degree