In this paper, we define labeled multigraphs with ports, a graph model which specifies connection points for nodes and allows multiple edges and loops. The dynamic evolution of these structures is expressed with multigraph rewrite rules and a multigraph rewriting relation. Then we encode the multigraphs and multigraph rewriting using algebraic terms and term rewriting to provide an operational ...

The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with...

A c-edge-colored multigraph has each edge colored with one of the c available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parame...

A c-edge-colored multigraph has each edge colored with one of the c available colors and no two parallel edges have the same color. A proper hamiltonian path is a path containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper hamiltonian path, depending on several parameter...

Given a c-edge-coloured multigraph, a proper Hamiltonian path is a path that contains all the vertices of the multigraph such that no two adjacent edges have the same colour. In this work we establish sufficient conditions for an edge-coloured multigraph to guarantee the existence of a proper Hamiltonian path, involving various parameters as the number of edges, the number of colours, the rainb...

We consider the colorings of the edges of a multigraph in such a way that every non-pendant vertex is incident to at least two edges of the same color. The maximum number of colors that can be used in such colorings is the upper chromatic index of a multigraph G, denoted by χ̄(G). We prove that if a multigraph G has n vertices, m edges, p pendant vertices and maximum c disjoint cycles, then χ̄(G)...

Proof of Lemma 1.1. Let H be the sub-multigraph of G consisting only of the vertices and the edges lying on at least one of the paths u and v. Then, the multigraph H is connected, and the paths u and v are two paths in H. Hence, there exist two distinct paths from x to y in H (namely, u and v). Notice that the multigraph H has at most k + ` edges (since its edges consist of the k edges of the p...

Abstract. We consider a preferential attachment process in which a multigraph is built one node at a time. The number of edges added at stage t, emanating from the new node, is given by some prescribed function f(t), generalising a model considered by Kleinberg and Kleinberg in 2005 where f was presumed constant. We show that if f(t) is asymptotically bounded above and below by linear functions...

Galvin ([7]) proved that every k-edge-colorable bipartite multigraph is kedge-choosable. Slivnik ([11]) gave a streamlined proof of Galvin's result. A multigraph G is said to be nearly bipartite if it contains a special vertex Vs such that G Vs is a bipartite multigraph. We use the technique in Slivnik's proof to obtain a list coloring analog of Vizing's theorem ([12]) for nearly bipartite mult...

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). The List-Colouring Conjecture (LCC) is that every multigraph G has edgechoosability (list chromatic index) ch’(G) equal to its chromatic index x’(G). In this paper the LCC is proved first for multicircuits, and then, building on results of Peterson and Woodall, for any multigraph G in which ...