We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp and we present all nonhamiltonian such graphs of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vert...

In a previous paper analytical equations were derived for the packing fraction of crystalline structures consisting of bimodal randomly placed hard spheres [H. J. H. Brouwers, Phys. Rev. E 76, 041304 (2007)]. The bimodal packing fraction was derived for the three crystalline cubic systems: viz., face-centered cubic, body-centered cubic, and simple cubic. These three equations appeared also to b...

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz’s theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric pro...

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is hamiltonian), by Thomassen (every 4-connected line graph is hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edgecoloring or a dominating cycle), which are known to be equivalent, are equivalent with the statement that every snark (i.e. a cyclically 4-edge-connec...

For a positive integer s, a graph Γ is called s-arc transitive if its full automorphism group AutΓ acts transitively on the set of s-arcs of Γ . Given a group G and a subset S of G with S = S−1 and 1 / ∈ S, let Γ = Cay(G, S) be the Cayley graph of G with respect to S and G R the set of right translations of G on G. Then G R forms a regular subgroup of AutΓ . A Cayley graph Γ = Cay(G, S) is call...

For a connected graph G of order |V (G)| ≥ 3 and a k-labelling c : E(G) → {1, 2, . . . , k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (l1, l2, . . . , lk), where li is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detect...

A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such that all other faces are 6-faces. We show that any fullerene graph on n vertices can be bipartized by removing O( √ n) edges. This bound is asymptotically optimal.

We prove that a graph admits a strongly 2-connected orientation if and only if it is 4-edge-connected, and every vertex-deleted subgraph is 2-edge-connected. In particular, every 4-connected graph has such an orientation while no cubic 3-connected graph has such an orientation.

A one-factorization of a regular graph G is perfect if the union of any two one-factors is a Hamiltonian cycle. A graph G is said to be P1F if it possess a perfect one-factorization. We prove that G is a P1F cubic graph if and only if L(G) is a P1F quartic graph. Moreover, we give some necessary conditions for the existence of a P1F planar graph.

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, we classify the cubic s-regular cubic graphs of order 14p for each s ≥ 1 and each prime p. © 2008 Elsevier B.V. All rights reserved.