# Algorithms and probabilistic bounds for the chromatic number of random composite graphs

## Abstract

"The composite graph coloring problem (CGCP) is a generalization of the standard graph coloring problem (SGCP). Associated with each vertex is a positive integer called its chromaticity. The chromaticity of a vertex specifies the number of consecutive colors which must be assigned to it.

An exact algorithm for solving the CGCP is presented. The algorithm is a generalization of the vertex-sequential with dynamic reordering approach for the SGCP. It is shown that the method is as effective on composite graphs as its counterpart is on standard graphs. Let π³ΜΜ(CGn,p) and π³ΜΜ(SGn,p) denote, respectively, the mean chromatic number of a sample of random composite and standard graphs of order n and edge density p. It is demonstrated that the ratio π³ΜΜ(CGn,p) / π³ΜΜ(SGn,p), depends on p, but, for fixed p, is essentially constant, over the range of values of n for which the algorithms were applied.

Several new heuristic methods for efficiently approximating π³(CGn,p) for large values of n are presented. Of these, the CDsatur and CDsaturI1 algorithms, which are generalizations of the well known Dsatur algorithm, are shown to be very competitive with previously tested procedures.

A known method for calculating probabilistic lower bounds for π³ΜΜ(SGn,p) is generalized to produce such bounds for π³ΜΜ(CGn,p). Also, a method for estimating the value of π³ΜΜ(SGn,p), is shown to produce probabilistic upper bounds for π³ΜΜ(SGn,p). This procedure is then generalized to a method for calculating probabilistic upper bounds for π³ΜΜ(CGn,p). The resulting bounds are used to evaluate the actual effectiveness of several heuristic algorithms. It is shown that, for fixed p, although the mean absolute error of the heuristic procedures appears to increase as n is varied from 100 to 1000, the mean relative error remains reasonably constant"--
Abstract, page iii.

Gillett, Billy E.

## Committee Member(s)

Ho, C. Y. (Chung You), 1933-1988
Prater, John Bruce, 1932-2002
Rigler, A. K.
Koederitz, Leonard

Computer Science

## Degree Name

Ph. D. in Computer Science

A report which is substantially this dissertation is available here for download.

## Publisher

University of Missouri--Rolla

Fall 1990

## Pagination

xv, 373 pages

Includes bibliographical references (pages 370-372).

## Document Type

Dissertation - Restricted Access

text

English

T 6129

## Print OCLC #

24125620

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