On hypohamiltonian snarks and a theorem of Fiorini

In 2003, Cavicchioli et al. corrected an omission in the statement and proof of Fiorini's theorem from 1983 on hypohamiltonian snarks. However, their version of this theorem contains an unattainable condition for certain cases. We discuss and extend the results of Fiorini and Cavicchioli et al. and present a version of this theorem which is more general in several ways. Using Fiorini's erroneous result, Steffen had shown that hypohamiltonian snarks exist for some orders n >= 10 and each even n >= 92. We rectify Steffen's proof by providing a correct demonstration of a technical lemma on flower snarks, which might be of separate interest. We then strengthen Steffen's theorem to the strongest possible form by determining all orders for which hypohamiltonian snarks exist. This also strengthens a result of Macajova and Skoviera. Finally, we verify a conjecture of Steffen on hypohamiltonian snarks up to 36 vertices

Generation and Properties of Snarks

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on $n\leq 36$ vertices. Previously lists up to $n=28$ vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated and typos corrected. This version differs from the published one in that the Arxiv-version has data about the automorphisms of snarks; Journal of Combinatorial Theory. Series B. 201

Hypohamiltonian and almost hypohamiltonian graphs

This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results

Constructing hypohamiltonian snarks with cyclic connectivity 5 and 6

A graph is hypohamiltonian if it is not hamiltonian but every vertex-deleted subgraph is. In this paper we study hypohamiltonian snarks – cubic hypohamiltonian graphs with chromatic index 4. We describe a method, based on superposition of snarks, which produces new hypohamiltonian snarks from old ones. By choosing suitable ingredients we can achieve that the resulting graphs are cyclically 5connected or 6-connected. Previously, only three sporadic hypohamiltonian snarks with cyclic connectivity 5 had been found, while the flower snarks of Isaacs were the only known family of cyclically 6-connected hypohamiltonian snarks. Our method produces hypohamiltonian snarks with cyclic connectivity 5 and 6 for all but finitely many even orders