In this paper we are interested in the fine-grained complexity of deciding whether there is a homomorphism from an input graph $G$ to a fixed graph $H$ (the $H$-Coloring problem). The starting point is that these problems can be viewed as constraint satisfaction problems (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 pPol($E_H$). For irreflexive graphs we observe that there is no pair of graphs $H$ and $H'$ such that pPol($E_H$) $\subseteq$ pPol($E_{H'}$), unless $E_{H'}= \emptyset$ or $H =H'$. More generally we show the existence of an $n$-ary relation $R$ whose partial polymorphisms strictly subsume those of $H$ and such that CSP($R$) is 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 nontrivial cliques, odd cycles, as well as certain cores, and we give an algebraic characterization of projective cores. As a by-product, we settle the Okrasa and Rz\k{a}\.zewski conjecture for all graphs of at most 7 vertices.

翻译：本文关注判定输入图$G$到固定图$H$是否存在同态（即$H$-着色问题）的细粒度复杂度。研究起点在于此类问题可视为约束满足问题（CSP），且二元关系的（部分）多态性对此类CSP复杂度分类研究至关重要。为此，我们首先探究二元对称关系$E_H$及其对应（部分）多态性pPol($E_H$)的表达能力。对于无反自环图，我们发现除非$E_{H'}= \emptyset$或$H =H'$，否则不存在图对$H$和$H'$使得pPol($E_H$) $\subseteq$ pPol($E_{H'}$)。更一般地，我们证明了存在一个$n$元关系$R$，其部分多态性严格包含$H$的部分多态性，且CSP($R$)是NP完全的当且仅当$H$包含长度不超过$n$的奇环。受此启发，我们还描述了非平凡团、奇环以及某些核的全多态性集合，并给出了射影核的代数刻画。作为副产品，我们解决了所有顶点数不超过7的图上的Okrasa和Rzążewski猜想。