The Heawood graph and K3;3 have the property that all of their 2-factors are Hamilton circuits. We call such graphs 2-factor hamiltonian. We prove that if G is a k-regular bipartite 2factor hamiltonian graph then either G is a circuit or k 1⁄4 3: Furthermore, we construct an infinite family of cubic bipartite 2-factor hamiltonian graphs based on the Heawood graph and K3;3 and conjecture that th...

We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac’s theorem on the existence of Hamiltonian cycles. Namely, we prove for 1 ≤ k ≤ n − 2 if G is a connected graph with A ⊂ V (G) such that dG(v) ≥ k for all v ∈ A, then there exists a subtree T of G such that V (T ) ⊃ A and dT (v) ≤ ⌈ n−1 k ⌉ for all v ∈ A.

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...

A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on n vertices with minimum degree at least n/2 is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac’s theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very st...

Despite the simplicity of the hexagonal graphene structure formed by carbon atoms, the electronic behavior shows fascinating properties, giving high expectation for the possible applications of graphene in the field. The Graphene Nano-Ribbon Field Effect Transistor (GNRFET) is an emerging technology that received much attention in recent years. In this paper, we investigate the device performan...

In this note we show that deciding the existence of a Hamiltonian cycle in a cubic plane graph is equivalent to the problem of the existence of an associated cubic plane multi-3-gonal graph with a Hamiltonian cycle which takes alternately left and right edges at each successive vertex, i.e. it is also a Petrie cycle. The Petrie Hamiltonian cycle in an n-vertex plane cubic graph can be recognize...

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...