304 research outputs found
S-Packing Colorings of Cubic Graphs
Given a non-decreasing sequence of positive
integers, an {\em -packing coloring} of a graph is a mapping from
to such that any two vertices with color
are at mutual distance greater than , . This paper
studies -packing colorings of (sub)cubic graphs. We prove that subcubic
graphs are -packing colorable and -packing
colorable. For subdivisions of subcubic graphs we derive sharper bounds, and we
provide an example of a cubic graph of order which is not
-packing colorable
Between proper and strong edge-colorings of subcubic graphs
In a proper edge-coloring the edges of every color form a matching. A
matching is induced if the end-vertices of its edges induce a matching. A
strong edge-coloring is an edge-coloring in which the edges of every color form
an induced matching. We consider intermediate types of edge-colorings, where
edges of some colors are allowed to form matchings, and the remaining form
induced matchings. Our research is motivated by the conjecture proposed in a
recent paper of Gastineau and Togni on S-packing edge-colorings (On S-packing
edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019), 63-75)
asserting that by allowing three additional induced matchings, one is able to
save one matching color. We prove that every graph with maximum degree 3 can be
decomposed into one matching and at most 8 induced matchings, and two matchings
and at most 5 induced matchings. We also show that if a graph is in class I,
the number of induced matchings can be decreased by one, hence confirming the
above-mentioned conjecture for class I graphs
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
Dichotomies properties on computational complexity of S-packing coloring problems
This work establishes the complexity class of several instances of the
S-packing coloring problem: for a graph G, a positive integer k and a non
decreasing list of integers S = (s\_1 , ..., s\_k ), G is S-colorable, if its
vertices can be partitioned into sets S\_i , i = 1,... , k, where each S\_i
being a s\_i -packing (a set of vertices at pairwise distance greater than
s\_i). For a list of three integers, a dichotomy between NP-complete problems
and polynomial time solvable problems is determined for subcubic graphs.
Moreover, for an unfixed size of list, the complexity of the S-packing coloring
problem is determined for several instances of the problem. These properties
are used in order to prove a dichotomy between NP-complete problems and
polynomial time solvable problems for lists of at most four integers
Generalized Colorings of Graphs
A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique
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