Improved algorithm to determine 3-colorability of graphs with the minimum degree at least 7

TL;DR

This paper presents improved algorithms with exponential complexity bounds for determining 3-colorability of graphs with minimum degree at least 7 or 8, advancing the efficiency of graph coloring decision procedures.

Contribution

It introduces new algorithms with refined exponential bounds for 3-colorability testing in graphs with minimum degree at least 7 or 8.

Findings

Algorithms with complexity $O(1.3158^{n-0.7~\Delta(G)})$ for $\delta(G)\geq 8$

Algorithms with complexity $O(1.32^{n-0.73~\Delta(G)})$ for $\delta(G)\geq 7$

Improved bounds enhance the efficiency of 3-colorability testing in graphs with high minimum degree.

Abstract

Let $G$ be an $n$-vertex graph with the maximum degree $\Delta$ and the minimum degree $\delta$. We give algorithms with complexity $O(1.3158^{n-0.7~\Delta(G)})$ and $O(1.32^{n-0.73~\Delta(G)})$ that determines if $G$ is 3-colorable, when $\delta(G)\geq 8$ and $\delta(G)\geq 7$, respectively.