Colouring Probe $H$-Free Graphs

TL;DR

This paper investigates the complexity of graph coloring problems in the probe graph model, providing a complete classification for 3-coloring on partitioned probe $P_t$-free graphs, revealing a complexity boundary at $t=6$.

Contribution

It introduces a new classification of coloring complexity in the probe graph model, highlighting differences from classical $H$-free graph results and establishing a dichotomy for 3-coloring.

Findings

3-Coloring is polynomial-time solvable for $t\, extleq 5$

3-Coloring is NP-complete for $t\, extgeq 6$

Contrast with classical $P_t$-free graphs where complexity thresholds differ

Abstract

The NP-complete problems Colouring and k-Colouring $(k\geq 3$) are well studied on $H$-free graphs, i.e., graphs that do not contain some fixed graph $H$ as an induced subgraph. We research to what extent the known polynomial-time algorithms for $H$-free graphs can be generalized if we only know some of the edges of the input graph. We do this by considering the classical probe graph model introduced in the early nineties. For a graph $H$, a partitioned probe $H$-free graph $(G,P,N)$ consists of a graph $G=(V,E)$, together with a set $P\subseteq V$ of probes and an independent set $N=V\setminus P$ of non-probes, such that $G+F$ is $H$-free for some edge set $F\subseteq \binom{N}{2}$. We first fully classify the complexity of Colouring on partitioned probe $H$-free graphs and show that this dichotomy is different from the known dichotomy of Colouring for $H$-free graphs. Our main result is a dichotomy of $3$-Colouring for partitioned probe $P_t$-free graphs: we prove that the problem is polynomial-time solvable if $t\leq 5$ but NP-complete if $t\geq 6$. In contrast, $3$-Colouring on $P_t$-free graphs is known to be polynomial-time solvable if $t\leq 7$ and quasi polynomial-time solvable for $t\geq 8$.