[Paper Review] Coloring $\Delta$-Critical Graphs With Small High Vertex Cliques

Landon Rabern|arXiv (Cornell University)|Feb 4, 2011

Advanced Graph Theory Research3 citations

TL;DR

This paper proves that for ∆-critical graphs with chromatic number χ(G) ≥ ∆(G) ≥ 6 and clique number ω(H(G)) ≤ ⌊∆(G)/2⌋ − 2, the only such graph is the complete graph Kχ(G). The proof uses a minimal coloring argument based on Kempe chain-like recoloring and structural analysis of high-degree vertices, resolving a conjecture by Kierstead and Kostochka that the bound 7 in their earlier result could be improved to 6.

ABSTRACT

We prove that $K_{\\chi(G)}$ is the only critical graph $G$ with $\\chi(G) \\geq \\Delta(G) \\geq 6$ and $\\omega(\\mathcal{H}(G)) \\leq \\left \\lfloor \\frac{\\Delta(G)}{2} \ ight \ floor - 2$. Here $\\mathcal{H}(G)$ is the subgraph of $G$ induced on the vertices of degree at least $\\chi(G)$. Setting $\\omega(\\mathcal{H}(G)) = 1$ proves a conjecture of Kierstead and Kostochka.

Motivation & Objective

• To resolve a conjecture by Kierstead and Kostochka that Kχ(G) is the only ∆-critical graph with χ(G) ≥ ∆(G) ≥ 6 when H(G) is independent.
• To generalize Brooks’ theorem using Ore-degree by showing that if χ(G) = ⌊θ(G)/2⌋ + 1 and χ(G) ≥ 6, then G contains a Kχ(G) clique.
• To establish structural constraints on ∆-critical graphs with small cliques in the subgraph H(G) induced by vertices of degree at least χ(G).
• To prove that under these constraints, no non-complete ∆-critical graph can exist, thereby characterizing the extremal case.

Proposed method

• Uses a minimal χ(G)-coloring of G that minimizes the number of edges in the union of color classes, based on an algorithmic approach inspired by Mozhan’s work.
• Applies a Kempe chain-like recoloring technique to swap vertices between color classes while preserving minimality, focusing on components induced by high-degree vertices.
• Analyzes the structure of components Z_i(x) induced by a vertex x and its assigned color class, showing they must be complete graphs or odd cycles under certain degree conditions.
• Employs induction on the number of vertices, assuming a minimal counterexample G ≠ Kχ(G) to derive contradictions via list coloring arguments.
• Applies Hall’s theorem to complete list colorings on subgraphs after removing cliques, ensuring colorability under degree and clique constraints.
• Uses a recursive vertex-swapping algorithm that tracks visit counts (q_i) to identify critical low-degree vertices and derive structural contradictions.

Experimental results

Research questions

• RQ1 Is Kχ(G) the only ∆-critical graph with χ(G) ≥ ∆(G) ≥ 6 and ω(H(G)) ≤ ⌊∆(G)/2⌋ − 2?
• RQ2 Can the bound 7 in the Kierstead-Kostochka conjecture be improved to 6 when H(G) is independent?
• RQ3 Under what conditions does a ∆-critical graph with χ(G) = ∆(G) contain a complete subgraph Kχ(G)?
• RQ4 Does the Ore-degree condition θ(G) = 2(χ(G)−1) imply the presence of a Kχ(G) clique in graphs with χ(G) ≥ 6?

Key findings

• Kχ(G) is the only ∆-critical graph with χ(G) ≥ ∆(G) ≥ 6 and ω(H(G)) ≤ ⌊∆(G)/2⌋ − 2, proving the main theorem.
• The conjecture of Kierstead and Kostochka is confirmed: Kχ(G) is the only ∆-critical graph with χ(G) ≥ ∆(G) ≥ 6 and H(G) independent.
• For ∆(G) = 6, G contains no K6 − e subgraph, as such a subgraph would allow a 5-coloring contradicting criticality.
• When ∆(G) = 6 and C is a K5 clique with at most one high vertex, each vertex outside C is adjacent to at most one low vertex in C.
• The recursive vertex-swapping algorithm ensures that q_k(y) ≤ 1 for all y ∈ V(G), which is essential for deriving contradictions in the coloring process.
• The contradiction arises when assuming a non-complete ∆-critical graph exists: the list coloring argument fails due to overlapping color lists and degree constraints, proving no such graph can exist.