## Dissertations

8-1989

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

Dr. Allen Schwenk

Dr. Jay Treiman

Dr. Gary Chartrand

Dr. Stephen Hedetniemi

#### Abstract

In Chapter I this concept is introduced. It is shown that computing the chromatic sum is NP-complete. For every natural k the smallest tree which needs k colors to attain its chromatic sum is constructed. It is demonstrated that asymptotically, for each k, almost all trees require more than k colors to achieve their chromatic sums. Also a linear algorithm for a single tree is presented.

In Chapter II three constructions of graphs that require t colors beyond their chromatic number k to achieve their chromatic sum are presented, depending on the ratio ${\rm t}\over{\rm k}$. The order of the resulting graphs grow linearly, quadratically, cubically and exponentially. The construction is proven to be the best possible for t = 1 and all k.

Chapter III deals with the chromatic sequences associated with a specific graph G, that is the sequence in k of the minimum sums of colors taken over all proper colorings of the graph G using exactly k colors. It is shown that for trees this sequence is constrained, in fact it is inverted unimodal, while for arbitrary graphs it is unconstrained.

In Chapter IV the weighted chromatic sum is investigated.

In Chapter V efficient algorithms for trees are presented. They are based on the Beyer - Hedetniemi and on Wright - Richmond - Odlyzko - McKay constant time algorithms generating all rooted trees and all free trees of a given order. Besides the generic algorithm, three specific algorithms are given: to find among all trees of a given order the tree with maximum average order of a subtree; to examine the frequencies of cospectral trees; and to examine the frequencies of the trees which need two colors to attain their chromatic sums. It is shown that, for rooted trees, the average number of steps per tree is bounded by a constant independent of the order of the trees. It is also conjectured that a similar property is true for free trees.

Appendices contain Pascal codes of the algorithms presented in Chapter V.