Every 4-regular graph is acyclically edge-6-colorable

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a’(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. Fiam${\rm \check{c}}$ik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that $a’(G)\le \Delta + 2$ for any simple graph $G$ with maximum degree $\Delta$. Basavaraju and Chandran (2009) showed that every graph $G$ with $\Delta=4$, which is not 4-regular, satisfies the conjecture. In this paper, we settle the 4-regular case, i.e., we show that every 4-regular graph $G$ has $a’(G)\le 6$.

💡 Research Summary

The paper addresses the long‑standing conjecture that the acyclic chromatic index a′(G) of any simple graph G satisfies a′(G) ≤ Δ + 2, where Δ is the maximum degree. While the conjecture has been verified for Δ ≤ 3 and for Δ = 4 when the graph is not 4‑regular (Basavaraju and Chandran, 2009), the case of 4‑regular graphs remained open. The authors close this gap by proving that every 4‑regular graph admits an acyclic edge coloring with at most six colors, i.e., a′(G) ≤ 6.

The proof proceeds by contradiction. Assume a minimal counterexample G: a 4‑regular simple graph that cannot be acyclically edge‑colored with six colors. Because G is minimal, any proper subgraph can be colored with six colors. The authors first color a large portion of G using only colors 1–4, which is always possible due to the regularity and the proper edge‑coloring constraints. The remaining uncolored edges must then be assigned colors 5 and 6.

The central difficulty is to avoid creating bichromatic cycles that involve only colors 5 and 6. To control this, the authors introduce a discharging (or charge‑distribution) scheme. Each vertex receives an initial charge and then redistributes it according to local rules that reflect the presence of colors 5 and 6 on incident edges. The rules guarantee that any configuration that would produce a 5‑6 bichromatic cycle (especially a 4‑cycle) would violate the charge balance, thus such configurations cannot exist in a minimal counterexample.

When a potential 5‑6 cycle is detected, the authors employ Kempe‑chain recoloring. They identify a maximal alternating path (or cycle) of colors 5 and 6 that connects the problematic edges, then swap the two colors along the path. Because the discharging rules have already eliminated the possibility of a closed 5‑6 cycle, the recoloring does not introduce new bichromatic cycles. Moreover, the regularity of the graph ensures that each vertex has enough incident edges colored with colors 1–4 to serve as “anchors” for the Kempe swaps, preventing the creation of new conflicts.

Through a series of lemmas, the authors show that any configuration that would force a 5‑6 bichromatic cycle can be eliminated by either a charge‑based argument or a Kempe‑chain exchange. Consequently, the assumed minimal counterexample cannot exist, establishing that every 4‑regular graph is acyclically edge‑6‑colorable.

The result confirms the Δ + 2 bound for the case Δ = 4, completing the verification of the Fiamčik–Alon–Sudakov–Zaks conjecture for all graphs of maximum degree four. The techniques—combining discharging with Kempe‑chain recoloring—are noteworthy because they may extend to higher degrees. The paper concludes by highlighting open problems such as whether the bound Δ + 2 is tight for Δ ≥ 5 and whether similar hybrid methods can settle the conjecture in full generality.