The topological approach to the study of infinite graphs of Diestel and K ̈hn has enabled several results on Hamilton cycles in finite graphs to be extended to locally finite graphs. We consider the result that the line graph of a finite 4-edge-connected graph is hamiltonian. We prove a weaker version of this result for infinite graphs: The line graph of locally finite, 6-edge-connected graph w...

Let G = (V, E) be a connected graph. Let G = (V, E) be a connected graph. An edge set F ⊂ E is said to be a k-restricted edge cut, if G− F is disconnected and every component of G− F has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is the cardinality of a minimum k-restricted edge cut of G. A graph G is λk-connected, if G contains a k-restricted edge cut. A λk...

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for ...

‎darafsheh and assari in [normal edge-transitive cayley graphs on non-abelian groups of order 4p‎, ‎where $p$ is a prime number‎, ‎sci‎. ‎china math‎., ‎56 (1) (2013) 213-219.] classified the connected normal edge transitive and‎ ‎$frac{1}{2}-$arc-transitive cayley graph of groups of order $4p$‎. ‎in this paper we continue this work by classifying the‎ ‎connected cayley graph of groups of order...

An edge (vertex) cut X of G is r-essential if G−X has two components each which at least r edges. A graph r-essentially k-edge-connected (resp. k-connected) it no vertex) cuts size less than k. If r=1, we simply call essential. Recently, Lai and Li proved that every m-edge-connected essentially h-edge-connected contains k edge-disjoint spanning trees, where k,m,h are positive integers such k+1≤...

Bounds on the minimum degree and on the number of vertices attaining it have been much studied for finite edge-/vertex-minimally kconnected/k-edge-connected graphs. We give an overview of the results known for finite graphs, and show that most of these carry over to infinite graphs if we consider ends of small degree as well as vertices.

We characterize all simple graphs such that each edge is a chord of some cycle. As a consequence, we characterize all simple 2-connected graphs such that, for any two adjacent vertices x and y, the local connectivity k(x, y) ≥ 3. We also make a conjecture about chords for 3-connected graphs.

We describe the relationship between the crossing number of a graph G with a 2-edge-cut C and the crossing numbers of the components of G− C. Let G be a connected graph with a 2-edge-cut C := [V1, V2]. Let u1u2, v1v2 be the edges of C, so that ui, vi ∈ Vi for i = 1, 2, and let Gi := G[Vi] and Gi := Gi +uivi. We show that if either G1 or G2 is not connected, then cr(G) = cr(G1) + cr(G2), and tha...

Graphs may be classified in many ways apart from their order. A complete graph has each distinct pair of vertices connected by an edge. Many interesting problems arise in looking at complete graphs, several of which have to do with determining whether or not it is possible to pass through every vertex and edge once and only once (a Peterson cycle). A connected graph allows movement from any ver...