A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping φ from V (G) to V (H), that is φ(x)φ(y) is an arc in H whenever xy is an arc in G. The oriented chromatic number of G is the minimum order of an oriented graph H such that G has a homomorphism to H . The oriented chromatic index of G is the minimum order of an oriented graph H such that the line-digraph ...

A series of recent papers shows that it is NP-complete to decide whether an oriented graph admits a homomorphism to the tournament T4 on 4 vertices containing a 4-circuit, each time on a smaller graph class. We improve these results by showing that homomorphism to T4 is NP-complete for bipartite planar subcubic graphs of arbitrarily large fixed girth. We also show that push homomorphism is NP-c...

List homomorphisms generalize list colourings in the following way: Given graphs G; H , and lists L(v) V (H); v 2 V (G), a list homomorphism of G to H with respect to the lists L is a mapping f : V (G) ! V (H) such that uv 2 E (G) implies f (u)f(v) 2 E (H), and f (v) 2 L(v) for all v 2 V (G). The list homomorphism problem for a xed graph H asks whether or not an input graph G together with list...

We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization V of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Joh...

This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We discuss vertex transitive graphs and Cayley graphs and their rather fundamental role in some asp...

We address the problem of cyclic termgraph rewriting. We propose a new framework where rewrite rules are tuples of the form (L,R, τ, σ) such that L and R are termgraphs representing the left-hand and the right-hand sides of the rule, τ is a mapping from the nodes of L to those of R and σ is a partial function from nodes of R to nodes of L. τ describes how incident edges of the nodes in L are co...

We show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this “fractal” property contributes to the spectacular properties of the homomorphism order. We first show the fractal property by using Sparse Incomparability Lemma and then by a more involved elementary argument.

Let H be a fixed graph without loops. We prove that if H is a co-circular arc bigraph then the minimum cost homomorphism problem to H admits a polynomial time constant ratio approximation algorithm; otherwise the minimum cost homomorphism problem to H is known to be not approximable. This solves a problem posed in an earlier paper. For the purposes of the approximation, we provide a new charact...

Provenance Graphs: Anticipating and Exploiting Schema-Level Data Provenance Daniel Zinn Bertram Ludäscher {dzinn,ludaesch}@ucdavis.edu Abstract. Provenance graphs capture flow and dependency information recorded during scientific workflow runs, which can be used subsequently to interpret, validate, and debug workflow results. In this paper, we propose a new concept, called abstract provenance g...

Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit c...