Faudree–Schelp–Gyárfás–Tuza conjecture on strong edge-coloring of subcubic graphs

Let $G$ be a graph with maximum degree $\Delta(G)=3$.

Faudree–Schelp–Gyárfás–Tuza conjecture. The following bounds hold:

1. $\chi'_s(G)\le 10$.
2. If $G$ is bipartite, then $\chi'_s(G)\le 9$.
3. If $G$ is planar, then $\chi'_s(G)\le 9$.
4. If $G$ is bipartite and for each edge $xy\in E(G)$ one has $d(x)+d(y)\le 5$, then $\chi'_s(G)\le 6$.
5. If $G$ is bipartite and has no 4-cycle, then $\chi'_s(G)\le 7$.
6. If $G$ is bipartite and its girth is large, then $\chi'_s(G)\le 5$.

The conjecture concerns strong edge-colorings, in which edges at distance at most 1 receive distinct colors; equivalently, it concerns proper vertex-colorings of the square of the line graph. The source states that four parts have been confirmed, while the counterexample presented in the paper resolves the remaining conjectural claims.