Meta-Cayley Graphs on Dihedral Groups

>Magister Scientiae - MScThe pursuit of graphs which are vertex-transitive and non-Cayley on groups has been ongoing for some time. There has long been evidence to suggest that such graphs are a very rarety in occurrence. Much success has been had in this regard with various approaches being used. The aim of this thesis is to find such a class of graphs. We will take an algebraic approach. We will define Cayley graphs on loops, these loops necessarily not being groups. Specifically, we will define meta-Cayley graphs, which are vertex-transitive by construction. The loops in question are defined as the semi-direct product of groups, one of the groups being Z₂ consistently, the other being in the class of dihedral groups. In order to prove non-Cayleyness on groups, we will need to fully determine the automorphism groups of these graphs. Determining the automorphism groups is at the crux of the matter. Once these groups are determined, we may then apply Sabidussi's theorem. The theorem states that a graph is Cayley on groups if and only if its automorphism group contains a subgroup which acts regularly on its vertex set.Chemicals Industries Education and Training Authority (CHIETA

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

Some meta-cayley graphs on dihedral groups

In this paper, we define meta-Cayley graphs on dihedral groups. We fully determine the automorphism groups of the constructed graphs in question. Further, we prove that some of the graphs that we have constructed do not admit subgroups which act regularly on their vertex set; thus proving that they cannot be represented as Cayley graphs on groups

Reducible 3-critical graphs

In this paper, we provide further insights into the reducibility of 3-critical graphs. A graph is 3-critical if it is class two, that is,has chromatic index 4, and the removal of any one edge renders a graph with chromatic index 3. We consider two types of reductions: the suppression of a 2-connected subgraph into a single edge; and the suppression of a 3-connected subgraph into a single vertex. That is, in cases where the 2- or 3-connected subgraph is cubic or strictly subcubic. We show that every cyclically 2-connected 3-critical graph can be reduced to a smaller 3-critical graph using this method. We also prove that every 3-critical graph which is cyclically k-connected with k = 2 or k = 3, contains a 3-critical minor which is cyclically k-connected with k≥ 4