Improved bounds on coloring of graphs

TL;DR

This paper establishes new upper bounds on various graph coloring parameters, including acyclic edge and vertex chromatic numbers, star-chromatic number, and frugal chromatic number, using an improved Lovász Local Lemma.

Contribution

The paper introduces tighter bounds for multiple graph coloring parameters by applying an enhanced version of the Lovász Local Lemma, advancing theoretical understanding.

Findings

Acyclic edge chromatic number bound:  9.62( -1)

Vertex chromatic number bound:  6.59  ^{4/3}+3.3 

Star-chromatic number bound:  4.34  ^{3/2}+1.5

Abstract

Given a graph $G$ with maximum degree $\Delta\ge 3$, we prove that the acyclic edge chromatic number $a'(G)$ of $G$ is such that $a'(G)\le\lceil 9.62 (\Delta-1)\rceil$. Moreover we prove that: $a'(G)\le \lceil 6.42(\Delta-1)\rceil$ if $G$ has girth $g\ge 5\,$; $a'(G)\le \lceil5.77 (\Delta-1)\rc$ if $G$ has girth $g\ge 7$; $a'(G)\le \lc4.52(\D-1)\rc$ if $g\ge 53$; $a'(G)\le \D+2\,$ if $g\ge \lceil25.84\D\log\D(1+ 4.1/\log\D)\rceil$. We further prove that the acyclic (vertex) chromatic number $a(G)$ of $G$ is such that $a(G)\le \lc 6.59 \Delta^{4/3}+3.3\D\rc$. We also prove that the star-chromatic number $\chi_s(G)$ of $G$ is such that $\chi_s(G)\le \lc4.34\Delta^{3/2}+ 1.5\D\rc$. We finally prove that the $\b$-frugal chromatic number $\chi^\b(G)$ of $G$ is such that $\chi^\b(G)\le \lc\max\{k_1(\b)\D,\; k_2(\b){\D^{1+1/\b}/ (\b!)^{1/\b}}\}\rc$, 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 Lov\'asz Local Lemma due to Bissacot, Fern\'andez, Procacci and Scoppola \cite{BFPS}.