Generation of cubic graphs

We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5

Generation of Cubic Graphs

We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5

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

Some infinite classes of asymmetric nearly Hamiltonian snarks

We determine the full automorphism group of each member of three infinite families of connected cubic graphs which are snarks. A graph is said to be nearly hamiltonian if it has a cycle which contains all vertices but one. We prove, in particular, that for every possible order n ≥ 28 there exists a nearly hamiltonian snark of order n with trivial automorphism group

Measurements of edge uncolourability in cubic graphs

Philosophiae Doctor - PhDThe history of the pursuit of uncolourable cubic graphs dates back more than a century. This pursuit has evolved from the slow discovery of individual uncolourable cubic graphs such as the famous Petersen graph and the Blanusa snarks, to discovering in nite classes of uncolourable cubic graphs such as the Louphekine and Goldberg snarks, to investigating parameters which measure the uncolourability of cubic graphs. These parameters include resistance, oddness and weak oddness, ow resistance, among others. In this thesis, we consider current ideas and problems regarding the uncolourability of cubic graphs, centering around these parameters. We introduce new ideas regarding the structural complexity of these graphs in question. In particular, we consider their 3-critical subgraphs, speci cally in relation to resistance. We further introduce new parameters which measure the uncolourability of cubic graphs, speci cally relating to their 3-critical subgraphs and various types of cubic graph reductions. This is also done with a view to identifying further problems of interest. This thesis also presents solutions and partial solutions to long-standing open conjectures relating in particular to oddness, weak oddness and resistance

Edge and total colourings of graphs

Die vorliegenden Arbeit enthält Ergebnisse zu Kanten- und Totalfärbungen von Graphen sowie verschiedenen Variationen dieser Färbungen. Eine Kantenfärbung eines Graphen G ist eine Zuordnung von Farben zu den Kanten von G, so dass adjazente Kanten unterschiedliche Farben erhalten. Eine Totalfärbung ist eine Färbung der Knoten und Kanten von G, so dass adjazente Knoten, adjazente Kanten sowie ein Knoten und eine inzidente Kante jeweils unterschiedlich gefärbt werden. Der chromatische Index bzw. die totalchromatische Zahl von G bezeichnen die kleinste Anzahl von Farben, mit denen G kantenfärbbar bzw. totalfärbbar ist. In dieser Arbeit wird unter anderem die totalchromatische Zahl zirkulanter Graphen mit Maximalgrad 3 bestimmt sowie ein Algorithmus entwickelt, der alle planaren kritischen Graphen der Kantenfärbung mit bis zu 12 Knoten konstruiert und darstellt. Das Konzept der Kreisfärbung von Graphen wird von Knoten- auf Kanten- und Totalfärbung übertragen; Eigenschaften des kreischromatischen Index und der kreistotalchromatischen Zahl werden bewiesen und exakte Werte für einige Graphenklassen ermittelt. Die listenchromatische Vermutung wird für outerplanare Graphen mit Maximalgrad >4 bewiesen. Die Konzepte der (a,b)- und (a,b,r)-Listen- färbung werden von Knotenfärbung auf Kantenfärbung übertragen; es werden Eigenschaften dieser Färbungen und Ergebnisse für einzelne Graphenklassen hergeleitet.This thesis contains results for edge and total colourings as well as for some variations of these colourings. An edge colouring of a graph G is an assignment of colours to the edges of G such that adjacent edges are coloured differently. A total colouring is a colouring of the vertices and edges of G such that adjacent vertices, adjacent edges as well as a vertex and an incident edge are coloured differently. The chromatic index or the total chromatic number of G denote the minimum number of colours such that G admits an edge colouring or a total colouring, respectively. Results in this thesis are - among others - the total chromatic number of circulant graphs with maximum degree 3 and an algorithm to construct and draw all planar critical graphs with at most 12 vertices. The concept of circular colourings is transferred from vertex to edge and total colourings. Properties of the circular chromatic index and the circular total chromatic number are proven and exact values are determined for some classes of graphs. The list chromatic conjecture is confirmed for outerplanar graphs with maximum degree >4; the concepts of (a,b)- and (a,b,r)-list colourings are transferred from vertex to edge colouring and properties of these colourings as well as results for special classes of graphs are given