Let G1 = (V1, E1) and G2 = (V2, E2) be two edge-coloured graphs (without multiple edges or loops). A homomorphism is a mapping φ : V1 7−→ V2 for which, for every pair of adjacent vertices u and v of G1, φ(u) and φ(v) are adjacent in G2 and the colour of the edge φ(u)φ(v) is the same as that of the edge uv. We prove a number of results asserting the existence of a graph G, edge-coloured from a s...

We study the following min-min random graph process G = (G0, G1, . . .): the initial state G0 is an empty graph on n vertices (n even). Further, GM+1 is obtained from GM by choosing a pair {v, w} of distinct vertices of minimum degree uniformly at random among all such pairs in GM and adding the edge {v, w}. The process may produce multiple edges. We show that GM is asymptotically almost surely...

The Berge–Sauer conjecture (see [2, 3]) says that any simple (no multiple edges and loops) 4-regular graph contains a 3-regular subgraph. This conjecture was proved in [4, 6]. In [1, 2] the Chevalley–Warning theorem was used to extend this result to graphs with multiple edges, which are 4-regular plus an edge. Our main result, Theorem 2.2, presents the sufficient condition for a 4-regular graph...

Throughout the paper, G denotes a graph that has no loops, but that may have multiple edges. The multi-set of edges of G is denoted by E(G). A graph G is simple if it has no (loops or) multiple edges. A biclique of G is a simple complete bipartite subgraph of G. A biclique decomposition of G is a collection of bicliques of G, such that each edge of G is in precisely one of the bicliques in the ...

We consider an elastic plate of infinite length and constant width supported simply along its two parallel edges and having a finite length crack along its centreline. In particular, we look for and find trapped modes (localised oscillations) in the presence of the crack. An explicit wide-spacing approximation based on the Wiener-Hopf technique applied to incident wave scattering by semi-infini...

We present a novel semi-local image descriptor which encodes multiple edges corresponding to the image structure boundaries around an interest point. The proposed method addresses the problem of poor edge detection through a robust, scale and orientation invariant, descriptor distance. In addition, a clustering of descriptors capable of extracting distinctive shapes from a set of descriptors is...

To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the Kac-Moody Lie algebra associated with the graph.

Let G = (V; E) be a /nite and undirected graph without loops and multiple edges. An edge is said to dominate itself and any edge adjacent to it. A subset D of E is called a perfect edge dominating set if every edge of E \ D is dominated by exactly one edge in D and an e cient edge dominating set if every edge of E is dominated by exactly one edge in D. The perfect (e cient) edge domination prob...