The strong chromatic index of a graph $G$, denoted by $\mathrm{sq}(G)$, is the minimum $k$ such that the edges of $G$ can be partitioned into $k$ sets of 'strongly independent' edges, that is, such that the subgraph of $G$ induced by each set is the union of vertex-disjoint edges.Is it true that, for any graph $G$ with maximum degree $\Delta$,$$\mathrm{sq}(G)\le \frac{5}{4}{\Delta }^2?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

What you are asking is exactly the **Erdős–Nešetřil strong edge-colouring (strong chromatic index) conjecture**: you want to split the edges into induced matchings (your “strongly independent edge sets”), and the question is whether this always needs at most (\tfrac54\Delta^2) parts. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.