Full-homomorphisms to paths and cycles

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.