A graph G is 1-Hamilton-connected if G − x is Hamilton-connected for every vertex x ∈ V (G). In the paper we introduce a closure concept for 1-Hamiltonconnectedness in claw-free graphs. If G is a (new) closure of a claw-free graph G, then G is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, G is the line graph of a multigraph, and for some x ∈ V (G), G − x is the line graph of a ...

Greedy Forwarding algorithm is a widely-used routing algorithm for wireless networks. However, it can fail if the wireless network topologies contain void scenarios (or simply called voids). Since Yao Graph and Theta Graph are two types of geometric graphs exploited to construct wireless network topologies, this paper studies whether these two types of graphs can contain voids. Specifically, th...

Let H be a family of connected graphs. A graph G is said to be H-free if G is H-free for every graph H in H. In [1] it was pointed that there is a family of connected graphs H not containing any induced subgraph of the claw having the property that the set of H-free connected graphs containing a claw is finite, provided also that those graphs have minimum degree at least two and maximum degree ...

Let τ(G) and τG(e) denote the number of spanning trees of a graph G and the number of spanning trees of G containing edge e of G, respectively. Ferrara, Gould, and Suffel asked if, for every rational 0 < p/q < 1 there existed a graph G with edge e ∈ E(G) such that τG(e)/τ(G) = p/q. In this note we provide constructions that show this is indeed the case. Moreover, we show this is true even if we...

It is proved that if G is a plane embedding of a K4-minor-free graph with maximum degree ∆, then G is entirely 7-choosable if ∆ ≤ 4 and G is entirely (∆+2)-choosable if ∆ ≥ 5; that is, if every vertex, edge and face of G is given a list of max{7,∆+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is pr...

A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. At first sight, there seem to be a great variety of types of claw-free graphs. For instance, there are line graphs, the graph of the icosahedron, complements of triangle-free graphs, and the Schläfli graph (an amazingly highly-symmetric graph with 27 vertices), and more; for instance, if we arrange vertices in a circle...

In this paper, we show that every 3-connected claw-free graph G has a 2-factor having at most max { 2 5(α+1), 1 } cycles, where α is the independence number of G. As a corollary of this result, we also prove that every 3-connected claw-free graph G has a 2-factor with at most ( 4|G| 5(δ+2)+ 2 5 ) cycles, where δ is the minimum degree of G. This is an extension of a known result on the number of...