an introduction to chromatic polynomials英文.pdf


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JOURNAL OF COMBINATORIALTHEORY 4, 52-71 (1968)
An Introduction to Chromatic Polynomials*
RONALD C. READ
Department of Mathematics, University of the West Indies, Kingston, Jamaica
Communicated by Frank Harary
ABSTRACT
This expository paper is a general introduction to the theory of chromatic pol-
ynomials. Chromatic polynomials are defined, their salient properties are derived, and
some practicaI methods for computing them are given. A brief mention is made of
the connection between the theory of chromatic polynomials and map coloring
problems. The paper concludes with some unsolved problems relating to chromatic
polynomials and to applications of the theory to practical problems in operations
research.
1. INTRODUCTION
There are many interesting problems which arise when one considers
the ways of coloring the nodes of a graph subject to certain restrictions.
The object of this paper is to give a brief account of the fundamentals
of this branch of graph theory.
A coloring of a graph is the result of giving to each node of the graph
one of a specified set of colors. In more mathematical terms it is a mapping
of the nodes into (or onto) a specified finite set C (the set of colors).
We shall leave for the moment the question of whether the mapping is
to be into or onto.
By a proper coloring of a graph will be meant a coloring which satisfies
the restriction that adjacent nodes are not given (., mapped onto) the
same color (element) of C. A coloring for which this is not true will be
called an improper coloring.
These are the definitions; but it so happens that we shall nearly always
be concerned with proper colorings only, and it will therefore be con-
venient to drop the term "proper" and agree that by "colorings" of a
graph we mean "proper colorings" unless the c

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