#### Date of Award

8-1982

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Gary Chartrand

#### Second Advisor

Dr. Arthur White

#### Third Advisor

Dr. Jim Stewart

#### Fourth Advisor

Dr. John Petro

#### Abstract

A factor is a spanning sub(di)graph of a (di)graph. Factors that are generated by an algorithm that incorporates an element of randomness are often called random factors. An isofactor is essentially a factor G that is either empty or for which there exists a connected regular (di)graph H having a nontrivial G-factorization. Several topics, each concerning random factors or isofactors, are investigated in this dissertation. An historical introduction to these topics is given in Chapter I.

In Chapter II we define an antipath and say that a digraph is randomly antitraceable if every nonspanning antipath can be extended (at its terminus) to a longer antipath and, therefore, to a hamiltonian antipath. We characterize randomly antitraceable digraphs.

Chapter III is devoted to randomly bitraceable graphs. A graph G is randomly bitraceable if there is a bifactorization G = R (CRPLUS) B such that any nonhamiltonian path whose edges alternate between R and B can be extended to a longer alternating path. We determine which randomly traceable graphs are randomly bitraceable and show that such graphs have unique bitraceable bifactorizations.

In Chapter IV we define randomly near-traceable graphs as a generalization of randomly traceable graphs; these are graphs that admit a special type of efficient depth-first search. We show that all complete multipartite graphs are randomly near-traceable. We also prove that the radius of a randomly near-traceable graph is 1 or 2.

In Chapter V we offer entirely graph-theoretic proofs that every graph and digraph is an isofactor. We show that for every nonempty graph G there is a connected regular G-factorable graph H with (chi)(H) = (chi)(G). We also show that for every asymmetric digraph D there is an asymmetric, regular, connected D-factorable digraph. With R(,0)(G) denoting the minimum degree of regularity among all regular connected G-factorable graphs, we show that, for T a tree, (DELTA)(T) (LESSTHEQ) r(,0)(T) (LESSTHEQ) (DELTA)(T) + 1 are sharp bounds. This and other parameters are evaluated for several important classes of graphs.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Fink, John Frederick, "Random Factors and Isofactors in Graphs and Digraphs" (1982). *Dissertations*. 2509.

http://scholarworks.wmich.edu/dissertations/2509

## Comments

Fifth Advisor: Dr. Linda Lesniak-Foster