On the equality of the partial Grundy and upper ochromatic numbers of graphs

• Published in: Discrete mathematics Vol. 272; no. 1; pp. 53 - 64
• Main Authors: Erdös, Paul, Hedetniemi, Stephen T., Laskar, Renu C., Prins, Geert C.E.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 28-10-2003; Elsevier
• Subjects: Achromatic number; Chromatic number; Colorings; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Graph theory; Grundy number; Mathematics; Sciences and techniques of general use; Grundy number; Colorings; Achromatic number; Chromatic number; Interpolation; Grundy coloration; Graph colouring; Dominance; Graph theory
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/S0012-365X(03)00184-5

A (proper)
k-
coloring of a graph
G is a partition
Π={
V
1,
V
2,…,
V
k
} of
V(
G) into
k independent sets, called
color classes. In a
k-coloring
Π, a vertex
v∈
V
i
is called a
Grundy vertex if
v is adjacent to at least one vertex in color class
V
j
, for every
j,
j<
i. A
k-coloring is called a
Grundy coloring if every vertex is a Grundy vertex. A
k-coloring is called a
partial Grundy coloring if every color class contains at least one Grundy vertex. In this paper we introduce partial Grundy colorings, and relate them to parsimonious proper colorings introduced by Simmons in 1982.