[Paper Review] Equitable Coloring of Graphs with Intermediate Maximum Degree

Bor-Liang Chen, Kuo‐Ching Huang+1 more|arXiv (Cornell University)|Aug 26, 2014

Advanced Graph Theory Research6 references3 citations

TL;DR

This paper proves that any graph G of order at least 6 with intermediate maximum degree Δ satisfying (|G|+1)/3 ≤ Δ < |G|/2 is equitably Δ-colorable, provided no component is a complete graph K_{Δ+1}. The result confirms a key case of Chen, Lih, and Wu's equitable coloring conjecture for graphs with Δ in this range, using structural analysis and degree-based contradiction arguments to establish equitable color class size balance.

ABSTRACT

If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote the number of vertices of $G$ and $Δ=Δ(G)$ the maximum degree of a vertex in $G$. We prove that a graph $G$ of order at least 6 is equitably $Δ$-colorable if $G$ satisfies $(|G|+1)/3 \leq Δ&lt; |G|/2$ and none of its components is a $K_{Δ+1}$.

Motivation & Objective

• To resolve a key case of Chen, Lih, and Wu's equitable coloring conjecture for graphs with intermediate maximum degree.
• To establish sufficient conditions under which a graph is equitably Δ-colorable when Δ is neither too small nor too large relative to |G|.
• To extend the known range of graphs for which equitable coloring with Δ colors is guaranteed, particularly in the intermediate Δ regime.
• To prove that the absence of K_{Δ+1} components is a sufficient condition for equitable Δ-colorability in this range.

Proposed method

• Structural analysis of maximal [r,s,t]-colorings with r+s+t = Δ+1 to derive contradictions under assumed non-equitable coloring.
• Use of degree constraints and neighborhood analysis to show that vertices must be fully connected in specific configurations to meet Δ-degree requirements.
• Application of claims about minimum edge counts between sets (e.g., ‖X_i, U_j‖ ≥ 1) to enforce structural consistency.
• Contradiction via the existence of disjoint 3-independent sets (B) or independent set partitions (A), violating the assumed maximality of the coloring.
• Leveraging the Hajnal–Szemerédi theorem and prior results on equitable coloring thresholds to bound the analysis.
• Proof by contradiction: assuming a maximal equitable coloring with Δ+1 classes leads to structural violations unless G contains a K_{Δ+1}, which is excluded.

Experimental results

Research questions

• RQ1 Under what conditions is a graph with intermediate maximum degree Δ equitably Δ-colorable?
• RQ2 Can the equitable coloring conjecture of Chen, Lih, and Wu be verified for graphs where (|G|+1)/3 ≤ Δ < |G|/2?
• RQ3 What structural constraints prevent equitable Δ-coloring in graphs with Δ in this intermediate range?
• RQ4 Is the absence of K_{Δ+1} components sufficient to guarantee equitable Δ-colorability for such graphs?

Key findings

• Any graph G of order at least 6 with (|G|+1)/3 ≤ Δ < |G|/2 and no component isomorphic to K_{Δ+1} is equitably Δ-colorable.
• The proof establishes that such graphs cannot admit a maximal [r,s,t]-coloring with r+s+t = Δ+1 without leading to a contradiction via the existence of disjoint 3-independent sets.
• The degree constraints force all vertices in key sets (e.g., X_i, U_j, v_j) to be fully connected in a way that violates the independence requirement unless K_{Δ+1} is present.
• The result confirms Chen, Lih, and Wu’s equitable coloring conjecture for the intermediate range of Δ, extending prior results for Δ ≥ |G|/2.
• The absence of K_{Δ+1} components is a necessary and sufficient structural condition for equitable Δ-colorability in this range.