Lovász and Plummer conjectured that there exist a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2cn perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than

We show that any 2−factor of a cubic graph can be extended to a maximum 3−edge-colorable subgraph. We also show that the sum of sizes of maximum 2− and 3−edge-colorable subgraphs of a cubic graph is at least twice of its number of vertices.

Tutte’s 5-Flow Conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. In 2004, Kochol proved that the conjecture is equivalent to its restriction on cyclically 6-edge connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero 5-flow.

It was conjectured by Hoffmann-Ostenhof that the edge set of every cubic graph can be decomposed into a spanning tree, a matching and a family of cycles. We prove the conjecture for 3-connected cubic plane graphs and 3-connected cubic graphs on the projective plane. Our proof provides a polynomial time algorithm to find the decomposition for 3-connected cubic plane graphs.

Integrity, a measure of network reliability, is de-ned as I(G) = min S⊂V {|S|+ m(G − S)}; where G is a graph with vertex set V and m(G− S) denotes the order of the largest component of G − S. We prove an upper bound of the following form on the integrity of any cubic graph with n vertices: I(G)¡ 3 n+ O (√ n ) : Moreover, there exist an in-nite family of connected cubic graphs whose integrity sa...

A special class of cubic graphs are the cycle permutation graphs. A cycle permutation graph Pn(α) is defined by taking two vertex-disjoint cycles on n vertices and adding a matching between the vertices of the two cycles.In this paper we determine a good upper bound for tenacity of cycle permutation graphs.

Consider a non-hamiltonian graph G. G is hypohamiltonian if for every vertex v in G, the graph G − v is hamiltonian. G is almost hypohamiltonian if there exists a vertex w in G such that G−w is non-hamiltonian, and G− v is hamiltonian for every vertex v 6= w. McKay asked in [J. Graph Theory, doi: 10.1002/jgt.22043] whether infinitely many planar cubic hypohamiltonian graphs of girth 5 exist. We...

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v1 and v2, and attaches the neighbors of v e...

Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs with a circuit missing only one vertex and bridgeless cubic graphs with a 2-factor consisting of two circuits. The first part of this result implies that Berge ...