Given two graphs G and H, there is a bi-resolving (or bi-covering) graph homomorphism from G to H if and only if their adjacency matrices satisfy certain matrix relations. We investigate the bi-covering extensions of bi-resolving homomorphisms and give several sufficient conditions for a bi-resolving homomorphism to have a bi-covering extension with an irreducible domain. Using these results, w...

We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries include efficient algorithms for computing weighted sums approximating the number of k-colorings and the number of independent sets in a graph, as well as a...

2 Preliminaries 2 2.1 Homomorphism numbers and densities . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Local convergence of a graph sequence . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Chromatic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Subtree counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Weight...

A colored mixed graph has vertices linked by both colored arcs and colored edges. The chromatic number of such a graph G is defined as the smallest order of a colored mixed graph H such that there exists a (arc-color preserving) homomorphism from G to H . We study in this paper the colored mixed chromatic number of planar graphs, partial 2-trees and outerplanar graphs with given girth.

We determine under which conditions certain natural models of random constraint satisfaction problems have sharp thresholds of satisfiability. These models include graph and hypergraph homomorphism, the (d, k, t)-model, and binary constraint satisfaction problems with domain size 3.

A signed graph (G,?) is a G along with function ?:E(G)?{+,?}. closed walk of positive (resp., negative) if it has an even odd) number negative edges, counting repetitions. homomorphism (simple) to another vertex-mapping that preserves adjacencies and signs walks. The chromatic the minimum vertices |V(H)| (H,?) which admits homomorphism. Homomorphisms graphs have been attracting growing attentio...

We consider a model of a random height function with long-range constraints on a discrete segment. This model was suggested by Benjamini, Yadin and Yehudayoff and is a generalization of simple random walk. The random function is uniformly sampled from all graph homomorphisms from the graph Pn,d to the integersZ, where the graph Pn,d is the discrete segment {0, 1, . . . , n} with edges between v...

We consider the entailment problem in the fragment of first-order logic (FOL) composed of existentially closed conjunctions of literals (without functions), denoted FOL(∃,∧,¬a). This problem can be recast as several fundamental problems in artificial intelligence and databases, namely query containment for conjunctive queries with negation, clause entailment for clauses without functions and qu...

A Weight graph is a connected (multi)graph with two vertices u and v of degree at least three and other vertices of degree two. Moreover, if any of these two vertices is removed, the remaining graph contains a cycle. A Weight graph is called simple if the degree of u and v is three. We show full computational complexity characterization of the problem of deciding the existence of a locally inje...