3-colorability of graphs with minimum degree at least 6

TL;DR

This paper presents an algorithm to determine 3-colorability of graphs with minimum degree at least 6 efficiently, using list coloring properties and complexity analysis.

Contribution

It introduces a new algorithm for testing 3-colorability in graphs with minimum degree 6, with improved exponential time complexity.

Findings

Determines $L$-choosability in $O(1.3196^{n_3+0.5n_2})$ time.

Concludes 3-colorability can be decided in $O(1.3196^{n-0.5 riangle(G)})$ time for graphs with minimum degree 6.

Provides complexity bounds based on list assignment and degree conditions.

Abstract

Let $G$ be an $n$-vertex graph and let $L:V(G)\rightarrow P(\{1,2,3\})$ be a list assignment over the vertices of $G$, where each vertex with list of size 3 and of degree at most 5 has at least three neighbors with lists of size 2. We can determine $L$-choosability of $G$ in $O(1.3196^{n_3+.5n_2})$ time, where $n_i$ is the number of vertices in $G$ with list of size $i$ for $i\in \{2,3\}$. As a corollary, we conclude that the 3-colorability of any graph $G$ with minimum degree at least 6 can be determined in $O(1.3196^{n-.5\Delta(G)})$ time.