Let G be a graph with odd edge-connectivity r. It is proved in this paper that if r > 3, then G has a 3-cycle (1, 2)-cover of total length at most ((r + 1)|E(G)|)/r . © 2005 Elsevier B.V. All rights reserved.

Given a connected graph $$G = (V, E)$$ , we seek to partition the vertex set V into k non-empty parts such that subgraph induced by each part is connected, and maximally balanced in way maximum cardinality of these minimized. We refer this problem as min-max denote it -BGP. The vertex-weighted version on trees has been studied since about four decades ago, which admits linear time exact algorit...

Article history: Received 17 January 2008 Available online 21 March 2012

Given a graph G on n vertices with maximal degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that if the edge connectivity k(G) satisfies k(G) À d/ log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

For an interger l > 1, the l-edge-connectivity λl(G) of G is defined to be the smallest number of edges whose removal leaves a graph with at least l components, if |V (G)| ≥ l; and λl(G) = |V (G)| if |V (G)| ≤ l. A graph G is (k, l)-edge-connected if the l-edge-connectivity of G is at least k. A sufficient and necessary condition for G to be minimally (k, k − 1)-edgeconnected is obtained in the...

A simplified and shortened proof is presented for a theorem of Jordán and Szigeti [2] on detachments preserving local edge-connectivity.

The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second ve...