Complexity of $C_k$-coloring in hereditary classes of graphs

TL;DR

This paper investigates the computational complexity of $C_k$-coloring problems within hereditary graph classes, providing polynomial-time algorithms for certain cases and NP-completeness results for others, depending on the structure of the forbidden subgraph $F$.

Contribution

It establishes polynomial algorithms for $C_k$-coloring in $P_9$-free graphs for odd $k eq 4$ and extends to list variants, while also proving NP-completeness for specific $F$-free graphs when certain subgraph conditions are met.

Findings

Polynomial-time algorithms for $C_k$-coloring in $P_9$-free graphs for odd $k eq 4$.

NP-completeness of $C_k$-coloring extension and list variants in certain $F$-free graphs.

Extension of algorithms to list coloring variants for even $k eq 4$.

Abstract

For a graph $F$, a graph $G$ is \emph{$F$-free} if it does not contain an induced subgraph isomorphic to $F$. For two graphs $G$ and $H$, an \emph{$H$-coloring} of $G$ is a mapping $f:V(G)\rightarrow V(H)$ such that for every edge $uv\in E(G)$ it holds that $f(u)f(v)\in E(H)$. We are interested in the complexity of the problem $H$-{\sc Coloring}, which asks for the existence of an $H$-coloring of an input graph $G$. In particular, we consider $H$-{\sc Coloring} of $F$-free graphs, where $F$ is a fixed graph and $H$ is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-{\sc Coloring} of $P_t$-free graphs. We show that for every odd $k \geq 5$ the $C_k$-{\sc Coloring} problem, even in the list variant, can be solved in polynomial time in $P_9$-free graphs. The algorithm extends for the case of list version of $C_k$-{\sc Coloring}, where $k$ is an even number of length at least 10. On the other hand, we prove that if some component of $F$ is not a subgraph of a subdividecd claw, then the following problems are NP-complete in $F$-free graphs: a)extension version of $C_k$-{\sc Coloring} for every odd $k \geq 5$, b) list version of $C_k$-{\sc Coloring} for every even $k \geq 6$.