Localized structure in graph decompositions




Bowditch, Flora Caroline

Journal Title

Journal ISSN

Volume Title



Let v ∈ Z+ and G be a simple graph. A G-decomposition of Kv is a collection F={F1,F2,...,Ft} of subgraphs of Kv such that every edge of Kv occurs in exactlyone of the subgraphs and every graph Fi ∈ F is isomorphic to G. A G-decomposition of Kv is called balanced if each vertex of Kv occurs in the same number of copies of G. In 2011, Dukes and Malloch provided an existence theory for balanced G-decompositions of Kv. Shortly afterwards, Bonisoli, Bonvicini, and Rinaldi introduced degree- and orbit-balanced G-decompositions. Similar to balanced decompositions,these two types of G-decompositions impose a local structure on the vertices of Kv. In this thesis, we will present an existence theory for degree- and orbit-balanced G-decompositions of Kv. To do this, we will first develop a theory for decomposing Kv into copies of G when G contains coloured loops. This will be followed by a brief discussion about the applications of such decompositions. Finally, we will explore anextension of this problem where Kv is decomposed into a family of graphs. We will examine the complications that arise with families of graphs and provide results for a few special cases.



design theory, graph theory, combinatorics, graph decompositions