Related papers: On Algorithms Based on Finitely Many Homomorphism …

We introduce new methods for understanding the topology of $\Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $\Hom(T,-)$ and $\Hom(-,G)$ as functors from…

A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every…

We extend the notion of graph homomorphism to cellularly embedded graphs (maps) by designing operations on vertices and edges that respect the surface topology; we thus obtain the first definition of map homomorphism that preserves both the…

The Hom complex $\mathrm{Hom}(G, H)$ of graphs is a simplicial complex associated to a pair of graphs $G$ and $H$, and its homotopy type is of interest in the graph coloring problem and the homomorphism reconfiguration problem. In this…

We examine ordered graphs, defined as graphs with linearly ordered vertices, from the perspective of homomorphisms (and colorings) and their complexities. We demonstrate the corresponding computational and parameterized complexities, along…

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum over all…

An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an…

A classical result by Lov\'asz asserts that two graphs $G$ and $H$ are isomorphic if and only if they have the same left profile, that is, for every graph $F$, the number of homomorphisms from $F$ to $G$ coincides with the number of…

We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper $n$-colourings are generalized to the new notion of distinguishing homomorphisms. We prove…

A homomorphism from a graph $X$ to a graph $Y$ is an adjacency preserving mapping $f:V(X) \rightarrow V(Y)$. We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph $X$ admits a…

We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any $n$-vertex, $d$-regular graph $G$ and any graph $H$ (possibly with loops), \[\hom(G,H) \leq \max\left\lbrace\hom(K_{d,d}, H)^{\frac{n}{2d}},…

A homomorphism $f$ from a guest graph $G$ to a host graph $H$ is locally bijective, injective or surjective if for every $u\in V(G)$, the restriction of $f$ to the neighbourhood of $u$ is bijective, injective or surjective, respectively.…

In this paper we design {\sf FPT}-algorithms for two parameterized problems. The first is \textsc{List Digraph Homomorphism}: given two digraphs $G$ and $H$ and a list of allowed vertices of $H$ for every vertex of $G$, the question is…

Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere. One of the most well studied problems in this area is…

We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph $H$ to $n$-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new…

For every $q\in \mathbb N$ let $\textrm{FO}_q$ denote the class of sentences of first-order logic FO of quantifier rank at most $q$. If a graph property can be defined in $\textrm{FO}_q$, then it can be decided in time $O(n^q)$. Thus,…

We consider the subgraph isomorphism problem where, given two graphs G (source graph) and F (pattern graph), one is to decide whether there is a (not necessarily induced) subgraph of G isomorphic to F. While many practical heuristic…

For a fixed graph $H$ and for arbitrarily large host graphs $G$, the number of homomorphisms from $H$ to $G$ and the number of subgraphs isomorphic to $H$ contained in $G$ have been extensively studied in extremal graph theory and graph…

Let $H$ be an undirected graph. In the List $H$-Homomorphism Problem, given an undirected graph $G$ with a list constraint $L(v) \subseteq V(H)$ for each variable $v \in V(G)$, the objective is to find a list $H$-homomorphism $f:V(G) \to…

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is…