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Acyclic Edge Coloring of 3-sparse Graphs

Nevil Anto, Manu Basavaraju, Shashanka Kulamarva

TL;DR

This paper proves Fiamčík's acyclic edge coloring conjecture for the class of $3$-sparse graphs, showing $a'(G) \le \Delta+2$ in general and a tighter bound $a'(G) \le \Delta+1$ when there exists an edge $xy$ with $d_G(x)+d_G(y) < \Delta+3$. The authors use a minimum counterexample argument combined with a $\Delta$-edge-degenerate structure, a careful deletion of an edge, and a comprehensive recoloring framework based on maximal $(\mu,\nu)$-bichromatic paths and color exchanges to extend colorings from $G'$ to $G$. The results also identify the remaining hard cases as bipartite graphs with partitions of degrees $3$ and $\Delta$, and establish that $\Delta+2$ colors suffice for all $3$-sparse graphs. The work advances the understanding of acyclic edge coloring in structured sparse graph classes and suggests algorithmic directions for efficient colorings in these families.

Abstract

A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamčík conjectured that for a graph $G$ with maximum degree $Δ$, $a'(G) \le Δ+2$. A graph $G$ is said to be $3$-sparse if every edge in $G$ is incident on at least one vertex of degree at most $3$. We prove the conjecture for the class of $3$-sparse graphs. Further, we give a stronger bound of $Δ+1$, if there exists an edge $xy$ in the graph with $d_G(x)+ d_G(y) < Δ+3$. When $ Δ> 3$, the $3$-sparse graphs where no such edge exists is the set of bipartite graphs where one partition has vertices with degree exactly $3$ and the other partition has vertices with degree exactly $Δ$.

Acyclic Edge Coloring of 3-sparse Graphs

TL;DR

This paper proves Fiamčík's acyclic edge coloring conjecture for the class of -sparse graphs, showing in general and a tighter bound when there exists an edge with . The authors use a minimum counterexample argument combined with a -edge-degenerate structure, a careful deletion of an edge, and a comprehensive recoloring framework based on maximal -bichromatic paths and color exchanges to extend colorings from to . The results also identify the remaining hard cases as bipartite graphs with partitions of degrees and , and establish that colors suffice for all -sparse graphs. The work advances the understanding of acyclic edge coloring in structured sparse graph classes and suggests algorithmic directions for efficient colorings in these families.

Abstract

A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph denoted by , is the minimum integer such that has an acyclic edge coloring with colors. Fiamčík conjectured that for a graph with maximum degree , . A graph is said to be -sparse if every edge in is incident on at least one vertex of degree at most . We prove the conjecture for the class of -sparse graphs. Further, we give a stronger bound of , if there exists an edge in the graph with . When , the -sparse graphs where no such edge exists is the set of bipartite graphs where one partition has vertices with degree exactly and the other partition has vertices with degree exactly .
Paper Structure (4 sections, 6 theorems, 2 figures)

This paper contains 4 sections, 6 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:ACI3sprs']}
  4. Conclusion

Key Result

Theorem 1

Let $G$ be a connected 3-sparse graph with maximum degree $\Delta$. If $G$ has an edge $xy$ with $d_G(x) + d_G(y) < \Delta + 3$, then $a'(G) \le \Delta+1$.

Figures (2)

  • Figure 1: Case 2.3. The neighbourhood of edge $xy$
  • Figure 2: Case b.2: There exists $y'$ in $N(y) \setminus y_1$ such that $g(yy') = \gamma$ for some $\gamma \in F_{xy}$, and exactly one of $\mu$, $\nu$ is present on $y'$.

Theorems & Definitions (20)

  • Conjecture 1: Fiamvcik1978AC
  • Theorem 1
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • Definition 1: Basavaraju2010AEC2deg
  • Definition 2: Basavaraju2010AEC2deg
  • Lemma 2: Basavaraju2010AEC2deg
  • Definition 3: Basavaraju2010AEC2deg
  • ...and 10 more