Better 3-coloring algorithms: excluding a triangle and a seven vertex path

TL;DR

This paper introduces a faster, simpler algorithm for list 3-coloring graphs without triangles or induced 7-vertex paths, and proves finiteness of minimal obstructions for certain graph classes, enabling polynomial-time certification.

Contribution

It provides a new efficient algorithm for list 3-coloring in ree graphs excluding triangles and P7, and characterizes cases with finitely many minimal obstructions.

Findings

Algorithm complexity improved to O(n^5(n+m)) for general graphs and O(n^2(n+m)) for bipartite graphs.

Finiteness of minimal obstructions established for P7, C3-free graphs, enabling polynomial certification.

Identifies other parameter cases with finite minimal obstructions for list k-coloring in P_t, C_ree graphs.

Abstract

We present an algorithm to color a graph $G$ with no triangle and no induced $7$-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this is a special case of the problem solved in [Combinatorica 38(4):779--801, 2018], that does not require the absence of triangles, the algorithm here is both faster and conceptually simpler. The complexity of the algorithm is $O(|V(G)|^5(|V(G)|+|E(G)|))$, and if $G$ is bipartite, it improves to $O(|V(G)|^2(|V(G)|+|E(G)|))$. Moreover, we prove that there are finitely many minimal obstructions to list 3-coloring $\{P_t,C_3\}$-free graphs if and only if $t \leq 7$. This implies the existence of a polynomial time certifying algorithm for list 3-coloring in $\{P_7,C_3\}$-free graphs. We furthermore determine other cases of $t, \ell$, and $k$ such that the family of minimal obstructions to list $k$-coloring in $\{P_t,C_{\ell}\}$-free graphs is finite.