A homomorphism from an oriented graph G to an oriented graph H is a mapping φ from the set of vertices of G to the set of vertices of H such that −−−−−→ φ(u)φ(v) is an arc in H whenever −→uv is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (Reca...

For digraphs D and H , a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed directed or undirected graph H and an input graph D, the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by ...

For digraphs D and H , a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed directed or undirected graph H and an input graph D, the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by ...

The oriented chromatic number of an oriented graph −→ G is the minimum order of an oriented graph −→ H such that −→ G admits a homomorphism to −→ H . The oriented chromatic number of an undirected graph G is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph G, defined as the m...

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 o...

Recently, the theory of dense graph limits has received attention from multiple disciplines including graph theory, computer science, statistical physics, probability, statistics, and group theory. In this paper we initiate the study of the general structure of differentiable graphon parameters F . We derive consistency conditions among the higher Gâteaux derivatives of F when restricted to the...

A (m,n)-colored mixed graph is a graph having arcs of m different colors and edges of n different colors. A graph homomorphism of a (m,n)-colored mixed graph G to a (m,n)colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is also an arc (edge) of color c. The (m,n)-colored mixed chromatic number χ(m,n)(G) of a (m,n)-colored mixed graph G, in...

Given a graph G, we investigate the problem of determining the parity of the number of homomorphisms from G to some other fixed graph H. We conjecture that this problem exhibits a complexity dichotomy, such that all parity graph homomorphism problems are either polynomial-time solvable or ⊕P–complete, and provide a conjectured characterisation of the easy cases. We show that the conjecture is t...

For a symmetric bounded measurable function W on [0, 1] and a simple graph F , the homomorphism density