In 1976, F. Loupekine created a method for constructing new snarks from already known ones. In the present work, we consider an infinite family of snarks constructed from the Petersen Graph using Loupekine’s method, and show that this family verifies Fulkerson’s Conjecture. In addition, we show that it is possible to extend this result to families constructed from snarks other than the Petersen...

The altitude of a graph G is the largest integer k such that for each linear ordering f of its edges, G has a (simple) path P of length k for which f increases along the edge sequence of P . We determine a necessary and sufficient condition for cubic graphs with girth at least five to have altitude three and show that for r 4, r-regular graphs with girth at least five have altitude at least fou...

We describe two new algorithms for the generation of all non-isomorphic cubic graphs with girth at least k ≥ 5 which are very efficient for 5 ≤ k ≤ 7 and show how these algorithms can be efficiently restricted to generate snarks with girth at least k. Our implementation of these algorithms is more than 30, respectively 40 times faster than the previously fastest generator for cubic graphs with ...

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...

A graph G is 1-Hamilton-connected if G − x is Hamilton-connected for every x ∈ V (G), and G is 2-edge-Hamilton-connected if the graph G + X has a hamiltonian cycle containing all edges of X for any X ⊂ E+(G) = {xy| x, y ∈ V (G)} with 1 ≤ |X| ≤ 2. We prove that Thomassen’s conjecture (every 4-connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent...

The Berge–Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is contained in exactly two of these perfect matchings. In this paper, a useful technical lemma is proved that a cubic graph G admits a Berge–Fulkerson coloring if and only if the graph G contains a pair of edge-disjoint matchings M1 and M2 such that (i) M1 ∪ M2 in...

We present preliminary work on an intelligent interface for answering English language clinical queries. Although our approach is domain independent, we focus on the needs of clinical researchers who are identifying cohorts of patients based on HIV drug-resistance patterns. Such questions are transformed into an unambiguous logical form by natural language technology (Bridge), which is then sen...