An Algebraic Approach Towards the Fine-Grained Complexity of Graph Coloring Problems

In this paper we are interested in the fine-grained complexity of determining whether there is an homomorphism from an input graph $G$ to a fixed graph $H$ (the H-coloring problem). The starting point is the observation that these problems can be formulated in the language of CSPs, and that (partial) polymorphisms of binary relations are of paramount importance in the study of complexity classes of such CSPs. Thus, we first investigate the expressivity of binary symmetric relations $E_{H}$ and their corresponding (partial) polymorphisms $\text{pPol}(E_{H})$. For irreflexive graphs we observe that there is no pair of graphs $H$ and $H^{\prime}$ such that $\text{pPol}(E_{H})\subseteq \text{pPol}(E_{H^{\prime}})$, unless $E_{H^{\prime}}=\emptyset$ or $H =H^{\prime}$. More generally we show the existence of an $n$-ary relation $R$ whose partial polymorphisms strictly subsume those of $H$ and such that $\text{CSP} (R)$ is $\mathcal{NP}$ complete if and only if $H$ contains an odd cycle of length at most $n$. Motivated by this we also describe the sets of total polymorphisms of every nontrivial clique, every odd cycle, as well as certain cores. We finish the paper with some noteworthy questions for future research.