Height function delocalisation on cubic planar graphs

Random cubic planar graphs

We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αnρn!, where ρ . = 3.13259 and α are analytic constants. We show also that the chromatic number of a random cubic planar graph that is chosen uniformly at random among all the labeled cubic planar graphs on n vertices is three with probability tending to e /4! . = 0.999568, and is four with probab...

On Cubic Planar Hypohamiltonian and Hypotraceable Graphs

We present a cubic planar hypohamiltonian graph on 70 vertices, improving the best known bound of 94 by Thomassen and derive some consequences concerning longest paths and cycles of planar 3-connected graphs. We also show that cubic planar hypohamiltonian graphs on n vertices exist for every even number n ≥ 86 and that cubic planar hypotraceable graphs exist on n vertices for every even number ...

Cubic Augmentation of Planar Graphs

In this paper we study the problem of augmenting a planar graph such that it becomes 3-regular and remains planar. We show that it is NP-hard to decide whether such an augmentation exists. On the other hand, we give an efficient algorithm for the variant of the problem where the input graph has a fixed planar (topological) embedding that has to be preserved by the augmentation. We further gener...

Random cubic planar graphs revisited

The planar cubic Cayley graphs

We obtain a complete description of the planar cubic Cayley graphs, providing an explicit presentation and embedding for each of them. This turns out to be a rich class, comprising several infinite families. We obtain counterexamples to conjectures of Mohar, Bonnington and Watkins. Our analysis makes the involved graphs accessible to computation, corroborating a conjecture of Droms. This thesis...