OPEN

Let $G$ be a graph of maximum degree $\Delta$. Is $G$ the union of at most $\tfrac{5}{4}\Delta^2$ sets of strongly independent edges (sets such that the induced subgraph is the union of vertex-disjoint edges)?

Asked by Erdős and Nešetřil in 1985 (see [FGST89]). This bound would be the best possible, as witnessed by a blowup of $C_5$. The minimum number of such sets required is sometimes called the strong chromatic index of $G$.

The weaker conjecture that there exists some $c>0$ such that $(2-c)\Delta^2$ sets suffice was proved by Molloy and Reed [MoRe97], who proved that $1.998\Delta^2$ sets suffice (for $\Delta$ sufficiently large). This was improved to $1.93\Delta^2$ by Bruhn and Joos [BrJo18] and to $1.835\Delta^2$ by Bonamy, Perrett, and Postle [BPP22].

Erdős and Nešetřil also asked the easier problem of whether $G$ containing at least $\tfrac{5}{4}\Delta^2$ many edges implies $G$ containing two strongly independent edges. This was proved independently by Chung-Trotter and Gyárfás-Tuza.