Affiliation:
1. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia + Institute of Mathematics, Physics and Mechanics, Ljubljana, Ljubljana, Slovenia + Faculty of Education, University of Maribor, Maribor, Slovenia
2. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia + Institute of Mathematics, Physics and Mechanics, Ljubljana, Ljubljana, Slovenia
Abstract
Let G be a graph and let S = (s1,s2,..., sk) be a non-decreasing
sequence of positive integers. An S-packing coloring of G is a mapping c :
V(G) ? {1, 2,..., k} with the following property: if c(u) = c(v) = i,
then d(u,v) > si for any i ? {1, 2,...,k}. In particular, if S = (1,
2, 3, ..., k), then S-packing coloring of G is well known under the name
packing coloring. Next, let r be a positive integer and u,v ? V(G). A
vertex u r-distance dominates a vertex v if dG(u, v)? r. In this paper, we
present a new concept of a coloring, namely distance dominator packing
coloring, defined as follows. A coloring c is a distance dominator packing
coloring of G if it is a packing coloring of G and for each x ? V(G) there
exists i ? {1,2, 3,...} such that x i-distance dominates each vertex
from the color class of color i. The smallest integer k such that there
exists a distance dominator packing coloring of G using k colors, is the
distance dominator packing chromatic number of G, denoted by ?d?(G). In
this paper, we provide some lower and upper bounds on the distance dominator
packing chromatic number, characterize graphs G with ?d?(G) ? {2,3}, and
provide the exact values of ?d?(G) when G is a complete graph, a star, a
wheel, a cycle or a path. In addition, we consider the relation between
?? (G) and ?d?(G) for a graph G.
Publisher
National Library of Serbia
Cited by
1 articles.
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