3,972 research outputs found
): Pg.282-287 JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access
ABSTRACT The packing chromatic number of a graph G is the smallest integer k for which there exists a mapping such that any two vertices of color i are at distance at least 1 i + . It is a frequency assignment problem used in wireless networks, which is also called broadcast coloring. It is proved that packing coloring is NP-complete for general graphs and even for trees. In this paper, we give the packing chromatic number for splitting of bi star graph, sierpiński graph, broken wheel, jahangir graph and 4 q P K
Parameterized Complexity of Equitable Coloring
A graph on vertices is equitably -colorable if it is -colorable and
every color is used either or times.
Such a problem appears to be considerably harder than vertex coloring, being
even for cographs and interval graphs.
In this work, we prove that it is for block
graphs and for disjoint union of split graphs when parameterized by the number
of colors; and for -free interval graphs
when parameterized by treewidth, number of colors and maximum degree,
generalizing a result by Fellows et al. (2014) through a much simpler
reduction.
Using a previous result due to Dominique de Werra (1985), we establish a
dichotomy for the complexity of equitable coloring of chordal graphs based on
the size of the largest induced star.
Finally, we show that \textsc{equitable coloring} is when
parameterized by the treewidth of the complement graph
On Packing Colorings of Distance Graphs
The {\em packing chromatic number} of a graph is the
least integer for which there exists a mapping from to
such that any two vertices of color are at distance at
least . This paper studies the packing chromatic number of infinite
distance graphs , i.e. graphs with the set of
integers as vertex set, with two distinct vertices being
adjacent if and only if . We present lower and upper bounds for
, showing that for finite , the packing
chromatic number is finite. Our main result concerns distance graphs with
for which we prove some upper bounds on their packing chromatic
numbers, the smaller ones being for :
if is odd and
if is even
Packing Chromatic Number of Distance Graphs
The packing chromatic number of a graph is the smallest
integer such that vertices of can be partitioned into disjoint classes
where vertices in have pairwise distance greater than
. We study the packing chromatic number of infinite distance graphs , i.e. graphs with the set of integers as vertex set and in which two
distinct vertices are adjacent if and only if . In
this paper we focus on distance graphs with . We improve some
results of Togni who initiated the study. It is shown that for sufficiently large odd and
for sufficiently large even . We also give a lower bound 12 for
and tighten several gaps for with small .Comment: 13 pages, 3 figure
Coloring translates and homothets of a convex body
We obtain improved upper bounds and new lower bounds on the chromatic number
as a linear function of the clique number, for the intersection graphs (and
their complements) of finite families of translates and homothets of a convex
body in \RR^n.Comment: 11 pages, 2 figure
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