On a conjecture of Foulds and Robinson about deltahedra

In a paper by Foulds and Robinson in Discrete Appl. Math. 1 (1979) 75-87, it is proved that any two deltahedra of the same order can be transformed into each other by a sequence of two kinds of operations defined in that paper. They also conjectured that one kind of the operations is redundant. In the present paper we prove that their conjecture is true

Decomposition of 3-connected cubic graphs

AbstractWe solve a conjecture of Foulds and Robinson (1979) on decomposable triangulations in the plane, in the more general context of a decomposition theory of cubic 3-connected graphs. The decomposition gives us a natural way to obtain some known results about specific homeomorphic subgraphs and the extremal diameter of 3-connected cubic graphs

Valid path-based graph vertex numbering

A labelling of a graph is an assignment of labels to its vertex or edge sets (or both), subject to certain conditions, a well established concept. A labelling of a graph G of order n is termed a numbering when the set of integers {1,...,n} is used to label the vertices of G distinctly. A 2-path (a path with three vertices) in a vertex-numbered graph is said to be valid if the number of its middle vertex is smaller than the numbers of its endpoints. The problem of finding a vertex numbering of a given graph that optimises the number of induced valid 2-paths is studied, which is conjectured to be in the NP-hard class. The reported results for several graph classes show that apparently there are not one or more numbering patterns applicable to different classes of graphs, which requires the development of a specific numbering for each graph class under study