We prove that a graph G of order n is p -arrangeable if its vertices can be ordered as v 1 , v 2 , ..., v n such that | N L i ( N Ri ( v i ))| ≤ p for each 1 ≤ i ≤ n − 1, where L i denotes the vertices of the graph and R i denotes its neighbors.Expand

Abstract. Let D be a simple digraph without loops or digons. For any $ v\in V(D) $, the first out-neighborhood N+(v) is the set of all vertices with out-distance 1 from v and the second neighborhood… Expand

Cauchy interlacing-type properties of the normalized Laplacian associated with the graph G, denoted ${\cal L}(G), are investigated, and the following result is established.Expand

Given a family of graphs F , a graph G is F-saturated if no element of F is a subgraph of G, but for any edge e in G, some element of F is a subgraph of G + e. Let sat(n,F) denote the minimum number… Expand

We show that if G is a graph such that every edge is in at least two triangles, then G contains a spanning tree with no vertex of degree 2 (a homeomorphically irreducible spanning tree).Expand

We address an old (1977) conjecture of a subset of the authors (a variant of Ryser's conjecture): in every r-coloring of the edges of a biclique [A,B] (complete bipartite graph), the vertex set can… Expand

We consider monochromatic subragraphs in two-colored graphs as guaranteed by Ramsey's theorem, and ask various questions concerning the degree in the two- colored complete graphs of vertices which are part of these subgraphs.Expand

In this work, we propose a framework with strategies that can transform almost any existing complete coverage algorithm with any coverage ratio to an algorithm that can -cover the area to trade for network lifetime.Expand