Let G = (V (G),E(G)) be a graph. and let g, f be two integer-valued functions defined on V (G) such that g(x)≤ f (x) for all x ∈V (G). G is called fractional (g, f )-covered if each edge e of G belongs to a fractional (g, f )-factor Gh such that h(e) = 1, where h is the indicator function of Gh. In this paper, sufficient conditions related to toughness and isolated toughness for a graph to be f...

─ Graph theory is atool for solving Capacitated Vehicle Routing Problem (CVRP). This paper describes the application of software based on graph theory to analyze and solve CVRP in a nationwide freight company.The analysis was performed on a fleet of 13 vehicles with similar technical specifications and load capacity (15 tons). Medellin and its metropolitan area municipalities were the origin ch...

This paper examines the competitiveness of winner-take-all price competition in homogeneous product oligopoly environments where underlying buyer demands and/or firms’ costs need not be continuous. Our analysis is motivated by the observation that a variety of economic settings have these features. For example, in 1996 an Ivy League university solicited bids from several vendors for its initiat...

Let G = (V (G), E(G)) be an undirected graph with a measure function μ assigning non-negative values to subgraphs H so that μ(H) does not exceed the clique cover number of H. When μ satisfies some additional natural conditions, we study the problem of separating G into two subgraphs, each with a measure of at most 2μ(G)/3 by removing a set of vertices that can be covered with a small number of ...

The stability number of a graph G, denoted by α(G), is the cardinality of a maximum stable set, and µ(G) is the cardinality of a maximum matching in G. If α(G) + µ(G) equals its order, then G is a König-Egerváry graph. In this paper we deal with square-stable graphs, i.e., the graphs G enjoying the equality α(G) = α(G 2), where G 2 denotes the second power of G. In particular, we show that a Kö...

A bipartite graph is said to be symmetric if it has symmetry of reflecting two vertex sets. This paper investigates matching structure of symmetric bipartite graphs. We first apply the Dulmage-Mendelsohn decomposition to a symmetric bipartite graph. The resulting components, which are matching-covered, turn out to have symmetry. We then decompose a matching-covered bipartite graph via an ear de...

A quasi-line graph is a graph in which the neighbourhood of any vertex can be covered by two cliques; every line graph is a quasi-line graph. Reed conjectured that for any graph G, χ(G) ≤ d12(∆(G) + 1 + ω(G))e [13]. We prove that the conjecture holds if G is a quasi-line graph, extending a result of King, Reed and Vetta, who proved the conjecture for line graphs [8], and improving the bound of ...

According to static models of industrial organization, a rise in competition decreases prices. In this paper, I test whether this conclusion can be reversed in the mobile telecommunications markets where dynamic e ciency e ects might be signi cant. The empirical test relies on the change in the intensity of competition introduced by the entry of the fourth mobile operator in France and the merg...

A linear forest is the union of a set of vertex disjoint paths. Akiyama, Exoo and Harary and independently Hilton have conjectured that the edges of every graph of maximum degree ∆ can be covered by d∆+1 2 e linear forests. We show that almost every graph can be covered with this number of linear forests. 1 Linear Forests, Directed Cycles, and the Probabilistic Method A linear forest is the uni...