In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks. This parameter can be related to significant topological and dynamic properties of the networks, including percolation, epidemic spreading, synchronization, and random walks. As Apollonian networks constitute an interesting family of maximal planar graphs which are simultaneously small-...

Spanning Tree Protocol (STP) is a link management standard that provides loop free paths in Ethernet networks. Deploying STP in metro area networks is inadequate because it does not meet the requirements of these networks. STP blocks redundant links, causing the risk of congestion close to the root. As a result, STP provides poor support for load balancing in metro Ethernet networks. A solution...

A spanning tree with no vertices of degree two is called a homeomorphically irreducible spanning tree (or a HIST ) of a graph. In [7], sets of forbidden subgraphs that imply the existence of a HIST in a connected graph of sufficiently large order were characterized. In this paper, we focus on characterizing connected P5-free graphs which have a HIST. As applications of our main result, we also ...

We show that the square of every connected S(K1,4)-free graph satisfying a matching condition has 2-connected spanning subgraph maximum degree at most 3. Furthermore, we characterise trees whose k. This generalises results on S(K1,3)-free graphs Henry and Vogler [7] Harary Schwenk [6], respectively.

A graph G is supereulerian if it has a spanning eulerian subgraph. We prove that every 3-edge-connected graph with the circumference at most 11 has a spanning eulerian subgraph if and only if it is not contractible to the Petersen graph. As applications, we determine collections F1, F2 and F3 of graphs to prove each of the following (i) Every 3-connected {K1,3, Z9}-free graph is hamiltonian if ...

A 2-factor in a graph is a spanning 2-regular subgraph, or equivalently a spanning collection of disjoint cycles. In this paper we investigate the existence of 2-factors with a bounded number of odd cycles in a graph. We extend results of Ryjáček, Saito, and Schelp (Closure, 2-factors, and cycle coverings in claw-free graphs, J. Graph Theory, 32 (1999), no. 2, 109-117) and show that the number ...