[Paper Review] Improved bounds on coloring of graphs

Sokol Ndreca, Aldo Procacci+1 more|arXiv (Cornell University)|May 11, 2010

Limits and Structures in Graph Theory15 references4 citations

TL;DR

This paper improves upper bounds on several graph coloring parameters—acyclic edge coloring, acyclic vertex coloring, star coloring, and frugal coloring—using an enhanced version of the Lovász Local Lemma. It establishes tighter bounds that depend on maximum degree $\Delta$ and girth $g$, showing $a'(G) \leq \lceil 4.52(\Delta-1)\rceil$ for $g \geq 53$ and $\chi^\beta(G) \leq \lceil \max\{k_1(\beta)\Delta,\, k_2(\beta)\Delta^{1+1/\beta}/(\beta!)^{1/\beta}\} \rceil$ with decreasing $k_1(\beta) \in [4,6]$, $k_2(\beta) \in [2,5]$.

ABSTRACT

Given a graph $G$ with maximum degree $Δ\ge 3$, we prove that the acyclic edge chromatic number $a'(G)$ of $G$ is such that $a'(G)\le\lceil 9.62 (Δ-1) ceil$. Moreover we prove that: $a'(G)\le \lceil 6.42(Δ-1) ceil$ if $G$ has girth $g\ge 5\,$; $a'(G)\le \lceil5.77 (Δ-1) c$ if $G$ has girth $g\ge 7$; $a'(G)\le \lc4.52(\D-1) c$ if $g\ge 53$; $a'(G)\le \D+2\,$ if $g\ge \lceil25.84\D\log\D(1+ 4.1/\log\D) ceil$. We further prove that the acyclic (vertex) chromatic number $a(G)$ of $G$ is such that $a(G)\le \lc 6.59 Δ^{4/3}+3.3\D c$. We also prove that the star-chromatic number $χ_s(G)$ of $G$ is such that $χ_s(G)\le \lc4.34Δ^{3/2}+ 1.5\D c$. We finally prove that the $\b$-frugal chromatic number $χ^\b(G)$ of $G$ is such that $χ^\b(G)\le \lc\max\{k_1(\b)\D,\; k_2(\b){\D^{1+1/\b}/ (\b!)^{1/\b}}\} c$, where $k_1(\b)$ and $k_2(\b)$ are decreasing functions of $\b$ such that $k_1(\b)\in[4, 6]$ and $k_2(\b)\in[2,5]$. To obtain these results we use an improved version of the Lovász Local Lemma due to Bissacot, Fernández, Procacci and Scoppola \cite{BFPS}.

Motivation & Objective

• To improve existing upper bounds on acyclic edge chromatic number $a'(G)$, acyclic vertex chromatic number $a(G)$, star chromatic number $\chi_s(G)$, and $\beta$-frugal chromatic number $\chi^\beta(G)$ for graphs with maximum degree $\Delta$.
• To refine these bounds by incorporating the girth $g$ of the graph, particularly for graphs with large girth.
• To apply an improved version of the Lovász Local Lemma based on cluster expansion and polymer gas models to derive tighter concentration bounds.
• To optimize constants in known coloring bounds, especially for $a'(G)$, by leveraging structural constraints such as girth and degree.

Proposed method

• Utilizes an enhanced version of the Lovász Local Lemma derived from cluster expansion and statistical mechanics, specifically the Bissacot-Fernandez-Procacci-Scoppola variant.
• Models unfavorable events for coloring constraints: adjacent vertices receiving same color and $\beta+1$ vertices in a neighborhood sharing the same color.
• Defines dependency graphs $H$ over edge and neighborhood sets, with cliques induced by shared vertices to bound dependencies.
• Assigns weight variables $\mu_1$, $\mu_2$ to edge and $\beta$-neighborhood events, optimizing over $\alpha = \Delta\mu_1$ to minimize required color count.
• Derives bounds via optimization of functions $k_1(\beta)$ and $k_2(\beta)$, which are decreasing in $\beta$, ensuring tighter asymptotic scaling.
• Applies Shearer's theorem and independent-set polynomial non-vanishing conditions to validate the Lovász Local Lemma conditions in the probabilistic framework.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound on the acyclic edge chromatic number $a'(G)$ for graphs with maximum degree $\Delta$ and girth $g$?
• RQ2 How do improved bounds on $a'(G)$ depend on girth, and what is the threshold girth for achieving $a'(G) \leq \Delta + 2$?
• RQ3 Can the $\beta$-frugal chromatic number $\chi^\beta(G)$ be bounded more tightly than previous results, especially for large $\beta$?
• RQ4 What is the best achievable upper bound on the acyclic vertex chromatic number $a(G)$ and star chromatic number $\chi_s(G)$ in terms of $\Delta$?
• RQ5 Can the constants in known coloring bounds be optimized using advanced probabilistic methods like the enhanced Lovász Local Lemma?

Key findings

• The acyclic edge chromatic number satisfies $a'(G) \leq \lceil 9.62(\Delta - 1) \rceil$ for any $\Delta \geq 3$, improving prior bounds.
• For graphs with girth $g \geq 5$, $a'(G) \leq \lceil 6.42(\Delta - 1) \rceil$; for $g \geq 7$, $a'(G) \leq \lceil 5.77(\Delta - 1) \rceil$; and for $g \geq 53$, $a'(G) \leq \lceil 4.52(\Delta - 1) \rceil$.
• When girth satisfies $g \geq \lceil 25.84\Delta\log\Delta(1 + 4.1/\log\Delta) \rceil$, $a'(G) \leq \Delta + 2$, confirming a conjecture for sufficiently large girth.
• The acyclic vertex chromatic number is bounded by $a(G) \leq \lceil 6.59\Delta^{4/3} + 3.3\Delta \rceil$, improving on the previous $50\Delta^{4/3}$ bound.
• The star chromatic number satisfies $\chi_s(G) \leq \lceil 4.34\Delta^{3/2} + 1.5\Delta \rceil$, improving the prior $20\Delta^{3/2}$ bound.
• The $\beta$-frugal chromatic number is bounded by $\chi^\beta(G) \leq \lceil \max\{k_1(\beta)\Delta,\, k_2(\beta)\Delta^{1+1/\beta}/(\beta!)^{1/\beta}\} \rceil$, where $k_1(\beta) \in [4,6]$ and $k_2(\beta) \in [2,5]$ are decreasing in $\beta$, yielding tighter asymptotic scaling.