Full-homomorphisms to paths and cycles

TL;DR

This paper characterizes all minimal obstructions for full-homomorphisms to paths and cycles, revealing their finite nature and quadratic bounds, and discusses open problems about their maximal sizes and counts.

Contribution

It provides a complete description of minimal obstructions for paths and cycles under full-homomorphisms, extending understanding of graph coloring constraints.

Findings

Finite number of minimal obstructions for paths and cycles.

Quadratic bounds on the number of minimal obstructions.

Open problems on maximal sizes and counts of minimal obstructions.

Abstract

A full-homomorphism between a pair of graphs is a vertex mapping that preserves adjacencies and non-adjacencies. For a fixed graph $H$, a full $H$-colouring is a full-homomorphism of $G$ to $H$. A minimal $H$-obstruction is a graph that does not admit a full $H$-colouring, such that every proper induced subgraph of $G$ admits a full $H$-colouring. Feder and Hell proved that for every graph $H$ there is a finite number of minimal $H$-obstructions. We begin this work by describing all minimal obstructions of paths. Then, we study minimal obstructions of regular graphs to propose a description of minimal obstructions of cycles. As a consequence of these results, we observe that for each path $P$ and each cycle $C$, the number of minimal $P$-obstructions and $C$-obstructions is $\mathcal{O}(|V(P)|^2)$ and $\mathcal{O}(|V(C)|^2)$, respectively. Finally, we propose some problems regarding the largest minimal $H$-obstructions, and the number of minimal $H$-obstructions.