P_4-Decomposability in Regular Graphs and Multigraphs

The main objective of this thesis is to review and expand the study of graph decomposability. An H-decomposition of a graph G=(V,E) is a partitioning of the edge set, $E$, into edge-disjoint isomorphic copies of a subgraph H. In particular we focus on the decompositions of graphs into paths. We prove that a 2,4 mutligraph with maximum multiplicity 2 admits a C_2,C_3-free Euler tour (and thus, a decomposition into paths of length 3 if it has size a multiple of 3) if and only if it avoids a set of 15 forbidden structures. We also prove that a 4-regular multigraph with maximum multiplicity 2 admits a decomposition into paths of length three if and only if it has size a multiple of 3 and no three vertices induce more than 4 edges. We go on to outline drafted work reflecting further research into path decomposition problems