#### Date of Award

12-1983

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Gary Chartrand

#### Second Advisor

Dr. Joseph Buckley

#### Third Advisor

Dr. Kung-Wei Yang

#### Abstract

A decomposition of a nonempty graph G is a collection of subgraphs G(,1), G(,2), ... , G(,k) of G such that their edge sets form a partition of the edge set of G. If G(,i) is isomorphic to a fixed graph H for each i, then G has an isomorphic decomposition into the graph H or, equivalently, G is H-decomposable. Several topics, each concerning isomorphic decompositions, are investigated in this dissertation.

An historical introduction to the subject of isomorphic decompositions is given in Chapter I. We also present there some new information on finding regular graphs that are H-decomposable for a prescribed graph H. These results are also extended to digraphs.

In order to illustrate more fully the main concepts of this dissertation, we study in Chapter II the problem of finding isomorphic decompositions of specific graphs, including the Petersen graph, for which all isomorphic decompositions are determined.

Chapter III is primarily devoted to isomorphic decompositions of complete and complete bipartite graphs into linear forests (graphs that are unions of paths). It is shown that if F is any linear forest of size n having no isolates, then K(,2n) is F-decomposable.

Chapter IV is concerned with decomposing complete graphs of prime order into vertex-symmetric (regular) graphs called circulants. Results obtained here give some bounds for the so-called isomorphic Ramsey numbers.

In Chapter V we define the concept of randomly H-decomposable graphs and characterize these graphs in two cases.

Previous results have determined extremal regular graphs that do not contain 1-factors and, in some instances, these graphs have been shown to possess a "near 1-factorization". Following in this direction, we find in Chapter VI those extremal regular graphs that fail to posses a near 1-factor. In a few cases we verify that these extremal graphs nevertheless contain a natural isomorphic factorization.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Ruiz, Sergio, "On Isomorphic Decompositions of Graphs" (1983). *Dissertations*. 2434.

https://scholarworks.wmich.edu/dissertations/2434