1,538 research outputs found
Notes on complexity of packing coloring
A packing -coloring for some integer of a graph is a mapping
such that any two vertices of color
are in distance at least . This concept
is motivated by frequency assignment problems. The \emph{packing chromatic
number} of is the smallest such that there exists a packing
-coloring of .
Fiala and Golovach showed that determining the packing chromatic number for
chordal graphs is \NP-complete for diameter exactly 5. While the problem is
easy to solve for diameter 2, we show \NP-completeness for any diameter at
least 3. Our reduction also shows that the packing chromatic number is hard to
approximate within for any .
In addition, we design an \FPT algorithm for interval graphs of bounded
diameter. This leads us to exploring the problem of finding a partial coloring
that maximizes the number of colored vertices.Comment: 9 pages, 2 figure
Packing Coloring of Undirected and Oriented Generalized Theta Graphs
The packing chromatic number (G) of an undirected (resp.
oriented) graph G is the smallest integer k such that its set of vertices V (G)
can be partitioned into k disjoint subsets V 1,..., V k, in such a way that
every two distinct vertices in V i are at distance (resp. directed distance)
greater than i in G for every i, 1 i k. The generalized theta graph
{\ell} 1,...,{\ell}p consists in two end-vertices joined by p 2
internally vertex-disjoint paths with respective lengths 1 {\ell} 1
. . . {\ell} p. We prove that the packing chromatic number of any
undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n
3 = |{i / 1 i p, {\ell} i = 3}|, and that both these bounds are
tight. We then characterize undirected generalized theta graphs with packing
chromatic number k for every k 3. We also prove that the packing
chromatic number of any oriented generalized theta graph lies between 2 and 5
and that both these bounds are tight.Comment: Revised version. Accepted for publication in Australas. J. Combi
Packing chromatic vertex-critical graphs
The packing chromatic number of a graph is the smallest
integer such that the vertex set of can be partitioned into sets ,
, where vertices in are pairwise at distance at least .
Packing chromatic vertex-critical graphs, -critical for short, are
introduced as the graphs for which
holds for every vertex of . If , then is
--critical. It is shown that if is -critical,
then the set can be almost
arbitrary. The --critical graphs are characterized, and
--critical graphs are characterized in the case when they
contain a cycle of length at least which is not congruent to modulo
. It is shown that for every integer there exists a
--critical tree and that a --critical
caterpillar exists if and only if . Cartesian products are also
considered and in particular it is proved that if and are
vertex-transitive graphs and , then is -critical
Generalizations of the Tree Packing Conjecture
The Gy\'arf\'as tree packing conjecture asserts that any set of trees with
vertices has an (edge-disjoint) packing into the complete graph
on vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some
special cases. We address the problem of packing trees into -chromatic
graphs. In particular, we prove that if all but three of the trees are stars
then they have a packing into any -chromatic graph. We also consider several
other generalizations of the conjecture
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 of two-step graphs: open packing partitioning of graphs
The two-step graphs are revisited by studying their chromatic numbers in this
paper. We observe that the problem of coloring of two-step graphs is equivalent
to the problem of vertex partitioning of graphs into open packing sets. With
this remark in mind, it can be considered as the open version of the well-known
-distance coloring problem as well as the dual version of total domatic
problem.
The minimum for which the two-step graph of a graph
admits a proper coloring assigning colors to the vertices is called the
open packing partition number of , that is,
p_{o}(G)=\chi\big{(}\mathcal{N}(G)\big{)}. We give some sharp lower and upper
bounds on this parameter as well as its exact value when dealing with some
families of graphs like trees. Relations between and some well-know
graph parameters have been investigated in this paper. We study this vertex
partitioning in the Cartesian, direct and lexicographic products of graphs. In
particular, we give an exact formula in the case of lexicographic product of
any two graphs. The NP-hardness of the problem of computing this parameter is
derived from the mentioned formula. Graphs for which equals the
clique number of are also investigated
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
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
- …