# Certifying coloring algorithms for graphs without long induced paths

## TL;DR

This paper proves finiteness results for minimal graphs without long induced paths or certain bipartite subgraphs that are not colorable in specific ways, leading to efficient certifying coloring algorithms.

## Contribution

It generalizes and extends previous results by establishing finiteness of minimal non-colorable graphs under certain induced subgraph constraints, enabling polynomial-time certifying algorithms.

## Key findings

- Finiteness of minimal graphs without induced $P_k$ and $K_{t,t}$ that are not $H$-colorable.
- Finiteness of minimal graphs without induced $P_k$ that are not $C_{k-2}$-colorable.
- Development of polynomial-time certifying algorithms for these coloring problems.

## Abstract

Let $P_k$ be a path, $C_k$ a cycle on $k$ vertices, and $K_{k,k}$ a complete bipartite graph with $k$ vertices on each side of the bipartition. We prove that (1) for any integers $k, t>0$ and a graph $H$ there are finitely many subgraph minimal graphs with no induced $P_k$ and $K_{t,t}$ that are not $H$-colorable and (2) for any integer $k>4$ there are finitely many subgraph minimal graphs with no induced $P_k$ that are not $C_{k-2}$-colorable. The former generalizes the result of Hell and Huang [Complexity of coloring graphs without paths and cycles, Discrete Appl. Math. 216: 211--232 (2017)] and the latter extends a result of Bruce, Hoang, and Sawada [A certifying algorithm for 3-colorability of $P_5$-Free Graphs, ISAAC 2009: 594--604]. Both our results lead to polynomial-time certifying algorithms for the corresponding coloring problems.