A graph G is a homomorphic preimage of another graph H, or equivalently G is H-colorable, if there exists a graph homomorphism f : G → H. A classic problem is to characterize the family of homomorphic preimages of a given graph H. A geometric graph G is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position A geometric homomorphism (resp. ...

A tropical graph (H, c) consists of a graph H and a (not necessarily proper) vertex-colouring c of H. For a fixed tropical graph (H, c), the decision problem (H, c)-Colouring asks whether a given input tropical graph (G, c1) admits a homomorphism to (H, c), that is, a standard graph homomorphism of G to H that also preserves vertex-colours. We initiate the study of the computational complexity ...

The oriented chromatic number o(H) of an oriented graph H is defined to be the minimum order of an oriented graph H ' such that H has a homomorphism to H' . If each graph in a class ~ has a homomorphism to the same H' , then H ' is ~-universal. Let ~k denote the class of orientations of planar graphs with girth at least k. Clearly, ~3 ~ ~4 ~ ~5... We discuss the existence of ~k-universal graphs...

Semantic consequence (entailment) in RDF is ususally computed using Pat Hayes Interpolation Lemma. In this paper, we reformulate this mechanism as a graph homomorphism known as projection in the conceptual graphs community. Though most of the paper is devoted to a detailed proof of this result, we discuss the immediate benefits of this reformulation: it is now easy to translate results from dif...

We introduce the maximum graph homomorphism (MGH) problem: given a graph G, and a target graph H, find a mapping φ : VG 7→ VH that maximizes the number of edges of G that are mapped to edges of H. This problem encodes various fundamental NP-hard problems including Maxcut and Max-k-cut. We also consider the multiway uncut problem. We are given a graphG and a set of terminals T ⊆ VG. We want to p...

The graph homomorphism problem (HOM) asks whether the vertices of a given n-vertex graph G can be mapped to the vertices of a given h-vertex graph H such that each edge of G is mapped to an edge of H. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the 2-CSP problem. In this paper, we prove several lower bounds for HOM under the Exponen...

For graphs G and H, a map f : V (G) 7→ V (H) is a homomorphism if f preserves adjacency. Let Hom(G,H) denote the set of all homomorphisms from G into H. In this paper, we proved that for a simple graph G with n = |V (G)| and for k with n ≥ k ≥ 5, if the odd girth of G is at least 2k+1 and if the minimum degree δ(G) > 2n/(2k+3), then Hom(G,Z2k+1) 6= ∅, where Z2k+1 denotes the cycle of length 2k ...

For a fixed graph $H$, the reconfiguration problem for $H$-colourings (i.e. homomorphisms to $H$) asks: given a graph $G$ and two $H$-colourings $\varphi$ and $\psi$ of $G$, does there exist a sequence $f_0,\dots,f_m$ of $H$-colourings such that $f_0=\varphi$, $f_m=\psi$ and $f_i(u)f_{i+1}(v)\in E(H)$ for every $0\leq i<m$ and $uv\in E(G)$? If the graph $G$ is loop-free, then this is the equiva...

We provide an upper bound to the number of graph homomorphisms from G to H, where H is a fixed graph with certain properties, and G varies over all N -vertex, d-regular graphs. This result generalizes a recently resolved conjecture of Alon and Kahn on the number of independent sets. We build on the work of Galvin and Tetali, who studied the number of graph homomorphisms from G to H when G is bi...

For given graphs G and H , let |Hom(G,H)| denote the set of graph homomorphisms from G to H . We show that for any finite, n-regular, bipartite graph G and any finite graph H (perhaps with loops), |Hom(G,H)| is maximum when G is a disjoint union of Kn,n’s. This generalizes a result of J. Kahn on the number of independent sets in a regular bipartite graph. We also give the asymptotics of the log...