We consider homomorphism properties of a random graph G(n, p), where p is a function of n. A core H is great if for all e ∈ E(H), there is some homomorphism from H−e to H that is not onto. Great cores arise in the study of uniquely H-colourable graphs, where two inequivalent definitions arise for general cores H. For a large range of p, we prove that with probability tending to 1 as n → ∞, G ∈ ...

The product homomorphism method is a combinatorial tool that can be used to develop polynomial PAC-learning algorithms in predicate logic. Using the product homomorphism method, we show that a single nonrecursive definite Horn clause is polynomially PAC-learnable if the background knowledge is a function-free extensional database over a single binary predicate and the ground atoms in the backgr...

This paper propagates the use of term graph rewriting systems as a formal computational model. The deenition of both term graphs and the reduction relation on term graphs are more complex than their corresponding notions in term rewriting systems. Consequently, it is often believed that reasoning about term graph rewriting systems is inherently more diicult than reasoning about term rewriting s...

We initiate a study of the homomorphism domination exponent of a pair of graphs F and G, defined as the maximum real number c such that |Hom(F, T )| > |Hom(G,T )| for every graph T . The problem of determining whether HDE(F,G) > 1 is known as the homomorphism domination problem and its decidability is an important open question arising in the theory of relational databases. We investigate the c...

In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant’s model. We are given a fixed graph H and want to find all graphs, from some graph class, homomorphic to this H. These graphs will be encoded by a family of polynomials. We give dichotomies for the polynomials for cycles, cliques, trees, outerplanar gra...

Let H be a fixed graph. We introduce the following list homomorphism problem: Given an input graph G and for each vertex v of G a ``list'' L(v) V(H), decide whether or not there is a homomorphism f : G H such that f (v) # L(v) for each v #V(G). We discuss this problem primarily in the context of reflexive graphs, i.e., graphs in which each vertex has a loop. We give a polynomial time algorithm ...

We prove that the coindex of the box complex B(H) of a graph H can be measured by the generalised Mycielski graphs which admit a homomorphism to it. As a consequence, we exhibit for every graph H a system of linear equations solvable in polynomial time, with the following properties: If the system has no solutions, then coind(B(H)) + 2 ≤ 3; if the system has solutions, then χ(H) ≥ 4.

In the course of extending Grötzsch’s theorem, we prove that every triangle-free graph without a K5-minor is 3-colorable. It has been recently proved that every triangle-free planar graph admits a homomorphism to the Clebsch graph. We also extend this result to the class of triangle-free graphs without a K5-minor. This is related to some conjectures which generalize the Four-Color Theorem. Whil...

Several possible definitions of local injectivity for a homomorphism an oriented graph G to H are considered. In each case, we determine the complexity deciding whether there exists such when is given and fixed tournament on three or fewer vertices. Each definition leads locally-injective colouring problem. A dichotomy theorem proved in case.

Graph homomorphisms from the $${\mathbb {Z}}^d$$ lattice to {Z}}$$ are functions on whose gradients equal one in absolute value. These height corresponding proper 3-colorings of and, two dimensions, 6-vertex model (square ice). We consider uniform model, obtained by sampling uniformly such a graph homomorphism subject boundary conditions. Our main result is that delocalizes having no translatio...