Burris–Schelp conjecture on the vertex-distinguishing chromatic index

Let $G$ be a graph with at most one isolated vertex and no isolated edge. For each degree $d$, let $n_d$ be the number of vertices of degree $d$, and define

$$\pi(G)=\min\left\{k\in\mathbb{N}\mathrel{\middle|}\binom{k}{d}\ge n_d\text{ for all }\delta(G)\le d\le\Delta(G)\right\}.$$

Here $\delta(G)$ and $\Delta(G)$ denote the minimum and maximum degrees of $G$, respectively, and $\chi'_{vd}(G)$ is the least number of colors in a vertex-distinguishing proper edge coloring of $G$.

Burris–Schelp conjecture.

$$\chi'_{vd}(G)\in\{\pi(G),\pi(G)+1\}.$$

The quantity $\pi(G)$ is a counting lower bound: vertices of degree $d$ must receive distinct $d$-element sets of incident colors. The conjecture predicts that this bound determines the vertex-distinguishing chromatic index up to an additive one for every graph satisfying the stated necessary conditions.