Fiamčík’s acyclic edge-coloring conjecture for arbitrary graphs

Prove that every finite simple graph with maximum degree Δ has acyclic chromatic index at most Δ+2.

Background

The paper studies acyclic edge coloring, in which a proper edge coloring contains no bichromatic cycle. The acyclic chromatic index a'(G) is the minimum number of colors in such a coloring. Fiamčík, independently with Alon, conjectured that the universal upper bound should be Δ+2, where Δ is the maximum degree of the graph.

The paper proves this conjectured bound for 3-sparse graphs, but explicitly notes that the conjecture remains unresolved for arbitrary graphs. It reports a substantially larger best-known general upper bound, emphasizing that the universal conjecture remains a challenging problem.

References

For a graph $G$ with maximum degree $\Delta$, $a'(G) \leq \Delta+2$. The conjecture remains unsolved for an arbitrary graph $G$, with the current best-known upper bound for $a'(G)$ being $3.569(\Delta-1)$, given by \citet{Fialho2020AECBound}.

— Acyclic Edge Coloring of 3-sparse Graphs  (2501.11281 - Anto et al., 20 Jan 2025) in Section 1, Introduction