Acyclic edge coloring of graphs

An {\em acyclic edge coloring} of a graph $G$ is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The {\em acyclic chromatic index} $\chiup_{a}’(G)$ of a graph $G$ is the least number of colors needed in an acyclic edge coloring of $G$. Fiam\v{c}'{i}k (1978) conjectured that $\chiup_{a}’(G) \leq \Delta(G) + 2$, where $\Delta(G)$ is the maximum degree of $G$. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC). A graph $G$ with maximum degree at most $\kappa$ is {\em $\kappa$-deletion-minimal} if $\chiup_{a}’(G) > \kappa$ and $\chiup_{a}’(H) \leq \kappa$ for every proper subgraph $H$ of $G$. The purpose of this paper is to provide many structural lemmas on $\kappa$-deletion-minimal graphs. By using the structural lemmas, we firstly prove that AECC is true for the graphs with maximum average degree less than four (\autoref{NMAD4}). We secondly prove that AECC is true for the planar graphs without triangles adjacent to cycles of length at most four, with an additional condition that every $5$-cycle has at most three edges contained in triangles (\autoref{NoAdjacent}), from which we can conclude some known results as corollaries. We thirdly prove that every planar graph $G$ without intersecting triangles satisfies $\chiup_{a}’(G) \leq \Delta(G) + 3$ (\autoref{NoIntersect}). Finally, we consider one extreme case and prove it: if $G$ is a graph with $\Delta(G) \geq 3$ and all the $3^{+}$-vertices are independent, then $\chiup_{a}’(G) = \Delta(G)$. We hope the structural lemmas will shed some light on the acyclic edge coloring problems.

💡 Research Summary

The paper investigates acyclic edge coloring, where an edge coloring is called acyclic if the subgraph induced by any two colors forms a linear forest (i.e., no bichromatic cycles). The central conjecture, known as the Acyclic Edge Coloring Conjecture (AECC), asserts that for any graph G the acyclic chromatic index χ′ₐ(G) is at most Δ(G)+2, where Δ(G) denotes the maximum degree. While the conjecture is known to hold for Δ≤3 and for several special families, the general case remains open.

The authors introduce a systematic study of κ‑deletion‑minimal graphs: graphs of maximum degree at most κ that require more than κ colors for an acyclic edge coloring, yet every proper subgraph can be colored with at most κ colors. By establishing a suite of structural lemmas about such minimal counterexamples, they obtain powerful local constraints on vertex degrees, adjacency patterns, and the interaction of color sets. Key tools include the notions of candidate and valid colors, (α,β)-alternating paths, (α,β,u,v)-critical paths, and two fundamental facts: (1) at most one maximal (α,β)-path can pass through a given vertex, and (2) in a κ‑deletion‑minimal graph any candidate color for a missing edge is invalid. Lemmas 1–4 show, for instance, that κ‑deletion‑minimal graphs are 2‑connected, that the sum of degrees of a vertex and its neighbors exceeds κ, and that any 2‑vertex forces its neighbors to have relatively high degree (≥ κ−Δ+4).

Armed with these lemmas, the authors prove four main results:

1. Theorem 4.3 (mad < 4). For any graph whose maximum average degree (mad) is less than 4, the conjectured bound holds: χ′ₐ(G) ≤ Δ(G)+2. The proof assumes a minimal counterexample, applies the structural lemmas to restrict its possible configurations, and then uses a discharging argument to show that such a configuration cannot exist, thereby confirming AECC for this broad class.

2. Theorem 4.4 (planar graphs without short‑cycle‑adjacent triangles). If a planar graph contains no triangle that shares an edge with a cycle of length ≤ 4, and every 5‑cycle contains at most three edges belonging to triangles, then again χ′ₐ(G) ≤ Δ(G)+2. The additional planar restrictions simplify the adjacency structure around high‑degree vertices, allowing the authors to extend the previous argument and obtain the exact AECC bound for this family. Several known results appear as corollaries.

3. Theorem 4.6 (planar graphs without intersecting triangles). When a planar graph has no pair of triangles that intersect (i.e., no two triangles share a vertex), the authors improve the best known general bound for such graphs from Δ+7 (or Δ+10) down to Δ+3. The proof again relies on the structural lemmas, but now exploits the absence of intersecting triangles to guarantee enough “color slack” around each high‑degree vertex, preventing the formation of bichromatic cycles after a careful recoloring process.

4. Extremal case (independent 3⁺‑vertices). If every vertex of degree at least 3 is independent (no two such vertices are adjacent), then the acyclic chromatic index equals the maximum degree: χ′ₐ(G)=Δ(G). In this situation the graph consists essentially of a high‑degree “core” surrounded by degree‑2 vertices, which eliminates the possibility of bichromatic cycles and enables a constructive Δ‑coloring.

Overall, the paper contributes a robust framework for attacking AECC via deletion‑minimal analysis. By translating global coloring constraints into precise local degree and adjacency conditions, the authors manage to settle the conjecture for several important graph families and to tighten existing bounds for planar graphs. The structural lemmas themselves are likely to be useful in future work aiming to push the AECC toward a full resolution.