Upper Bounds on the Acyclic Chromatic Index of Degenerate Graphs

TL;DR

The paper tackles the problem of bounding the acyclic chromatic index a'(G) for degenerate graphs, a central issue in acyclic edge coloring. It develops an inductive coloring framework that uses color exchanges and maximal bichromatic paths to extend colorings from subgraphs to the original graph. The authors prove two main bounds: for any k-degenerate graph with maximum degree Δ, a'(G) ≤ ceil(((k+1)/2) Δ) + 1, and specifically for 3-degenerate graphs, a'(G) ≤ Δ+5. These results move the known bounds closer to the conjectured Δ+2 and offer techniques that may extend to broader classes of degenerate graphs.

Abstract

An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamčík conjectured that $a'(G) \le Δ+2$ for any graph $G$ with maximum degree $Δ$. A graph $G$ is said to be $k$-degenerate if every subgraph of $G$ has a vertex of degree at most $k$. Basavaraju and Chandran proved that the conjecture is true for $2$-degenerate graphs. We prove that for a $3$-degenerate graph $G$, $a'(G) \le Δ+5$, thereby bringing the upper bound closer to the conjectured bound. We also consider $k$-degenerate graphs with $k \ge 4$ and give an upper bound for the acyclic chromatic index of the same.

Figures (3)

• Figure 1: Neighborhood of the edge $xy$ in $G$
• Figure 2: Neighborhood of the edge $xy$ in $G$ in Case \ref{['case:AlphaNotInSDegX2']}
• Figure 3: Neighborhood of the edge $xy$ in $G$ in Case \ref{['case:OneColorInFxy']}

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Definition 1: Basavaraju2010AEC2deg
• Definition 2: Basavaraju2010AEC2deg
• Lemma 1: Basavaraju2010AEC2deg
• Definition 3: Basavaraju2010AEC2deg
• Definition 4: Basavaraju2010AEC2deg
• Lemma 2: Basavaraju2010AEC2deg
• Lemma 3
• proof