378 research outputs found
Bounded Max-Colorings of Graphs
In a bounded max-coloring of a vertex/edge weighted graph, each color class
is of cardinality at most and of weight equal to the weight of the heaviest
vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask
for such a coloring minimizing the sum of all color classes' weights.
In this paper we present complexity results and approximation algorithms for
those problems on general graphs, bipartite graphs and trees. We first show
that both problems are polynomial for trees, when the number of colors is
fixed, and approximable for general graphs, when the bound is fixed.
For the bounded max-vertex-coloring problem, we show a 17/11-approximation
algorithm for bipartite graphs, a PTAS for trees as well as for bipartite
graphs when is fixed. For unit weights, we show that the known 4/3 lower
bound for bipartite graphs is tight by providing a simple 4/3 approximation
algorithm. For the bounded max-edge-coloring problem, we prove approximation
factors of , for general graphs, , for
bipartite graphs, and 2, for trees. Furthermore, we show that this problem is
NP-complete even for trees. This is the first complexity result for
max-coloring problems on trees.Comment: 13 pages, 5 figure
Graph coloring with no large monochromatic components
For a graph G and an integer t we let mcc_t(G) be the smallest m such that
there exists a coloring of the vertices of G by t colors with no monochromatic
connected subgraph having more than m vertices. Let F be any nontrivial
minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any
n-vertex graph G \in F. This bound is asymptotically optimal and it is attained
for planar graphs. More generally, for every such F and every fixed t we show
that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G
with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}).
It is also interesting to consider graphs of bounded degrees. Haxell, Szabo,
and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5.
We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n),
and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex
graphs of maximum degree 7, average degree at most 6+\epsilon for all
subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only
that the maximum order of magnitude of \mcc_2 is between \sqrt n and n.
We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).Comment: 13 pages, 2 figure
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
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
On the approximability of the maximum induced matching problem
In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3
Total Domination, Separated Clusters, CD-Coloring: Algorithms and Hardness
Domination and coloring are two classic problems in graph theory. The major
focus of this paper is the CD-COLORING problem which combines the flavours of
domination and colouring. Let be an undirected graph. A proper vertex
coloring of is a if each color class has a dominating vertex
in . The minimum integer for which there exists a of
using colors is called the cd-chromatic number, . A set
is a total dominating set if any vertex in has a neighbor
in . The total domination number, of is the minimum
integer such that has a total dominating set of size . A set
is a if no two vertices in lie at a
distance 2 in . The separated-cluster number, , of is the
maximum integer such that has a separated-cluster of size .
In this paper, first we explore the connection between CD-COLORING and TOTAL
DOMINATION. We prove that CD-COLORING and TOTAL DOMINATION are NP-Complete on
triangle-free -regular graphs for each fixed integer . We also
study the relationship between the parameters and .
Analogous to the well-known notion of `perfectness', here we introduce the
notion of `cd-perfectness'. We prove a sufficient condition for a graph to
be cd-perfect (i.e. , for any induced subgraph
of ) which is also necessary for certain graph classes (like triangle-free
graphs). Here, we propose a generalized framework via which we obtain several
exciting consequences in the algorithmic complexities of special graph classes.
In addition, we settle an open problem by showing that the SEPARATED-CLUSTER is
polynomially solvable for interval graphs
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