Faudree–Gyárfás–Schelp–Tuza strong clique index conjecture

For a finite simple graph $G$, let $L(G)$ denote its line graph, let $G^2$ denote the graph in which two vertices are adjacent exactly when they are at distance at most two in $G$, let $\omega$ denote clique number, and let $\Delta(G)$ denote the maximum degree of $G$. Faudree–Gyárfás–Schelp–Tuza conjecture. For any graph $G$,

$$\omega(L(G)^2) \le \frac{5}{4}\Delta(G)^2.$$

This is a weaker counterpart to the Erdős–Nešetřil conjecture for the strong chromatic index. It is tight for blow-ups of $C_5$ and remains open; the best-known general upper bound stated in the paper is $\frac{4}{3}\Delta(G)^2$.