9,498 research outputs found
Vertex-Coloring 2-Edge-Weighting of Graphs
A -{\it edge-weighting} of a graph is an assignment of an integer
weight, , to each edge . An edge weighting naturally
induces a vertex coloring by defining for every
. A -edge-weighting of a graph is \emph{vertex-coloring} if
the induced coloring is proper, i.e., for any edge .
Given a graph and a vertex coloring , does there exist an
edge-weighting such that the induced vertex coloring is ? We investigate
this problem by considering edge-weightings defined on an abelian group.
It was proved that every 3-colorable graph admits a vertex-coloring
-edge-weighting \cite{KLT}. Does every 2-colorable graph (i.e., bipartite
graphs) admit a vertex-coloring 2-edge-weighting? We obtain several simple
sufficient conditions for graphs to be vertex-coloring 2-edge-weighting. In
particular, we show that 3-connected bipartite graphs admit vertex-coloring
2-edge-weighting
Interval non-edge-colorable bipartite graphs and multigraphs
An edge-coloring of a graph with colors is called an interval
-coloring if all colors are used, and the colors of edges incident to any
vertex of are distinct and form an interval of integers. In 1991 Erd\H{o}s
constructed a bipartite graph with 27 vertices and maximum degree 13 which has
no interval coloring. Erd\H{o}s's counterexample is the smallest (in a sense of
maximum degree) known bipartite graph which is not interval colorable. On the
other hand, in 1992 Hansen showed that all bipartite graphs with maximum degree
at most 3 have an interval coloring. In this paper we give some methods for
constructing of interval non-edge-colorable bipartite graphs. In particular, by
these methods, we construct three bipartite graphs which have no interval
coloring, contain 20,19,21 vertices and have maximum degree 11,12,13,
respectively. This partially answers a question that arose in [T.R. Jensen, B.
Toft, Graph coloring problems, Wiley Interscience Series in Discrete
Mathematics and Optimization, 1995, p. 204]. We also consider similar problems
for bipartite multigraphs.Comment: 18 pages, 7 figure
Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights
Let be a graph and be a subset of . A vertex-coloring
-edge-weighting of is an assignment of weight by the
elements of to each edge of so that adjacent vertices have
different sums of incident edges weights.
It was proved that every 3-connected bipartite graph admits a vertex-coloring
-edge-weighting (Lu, Yu and Zhang, (2011) \cite{LYZ}). In this paper,
we show that the following result: if a 3-edge-connected bipartite graph
with minimum degree contains a vertex such that
and is connected, then admits a vertex-coloring
-edge-weighting for . In
particular, we show that every 2-connected and 3-edge-connected bipartite graph
admits a vertex-coloring -edge-weighting for . The bound is sharp, since there exists a family of
infinite bipartite graphs which are 2-connected and do not admit
vertex-coloring -edge-weightings or vertex-coloring
-edge-weightings.Comment: In this paper, we show that every 2-connected and 3-edge-connected
bipartite graph admits a vertex-coloring S-edge-weighting for S\in
{{0,1},{1,2}
Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees
An edge coloring of a graph G is called interval edge coloring if for each v ? V(G) the set of colors on edges incident to v forms an interval of integers. A graph G is interval colorable if there is an interval coloring of G. For an interval colorable graph G, by the interval chromatic index of G, denoted by ?\u27_i(G), we mean the smallest number k such that G is interval colorable with k colors. A bipartite graph G is called (?,?)-biregular if each vertex in one part has degree ? and each vertex in the other part has degree ?. A graph G is called (?*,?*)-bipartite if G is a subgraph of an (?,?)-biregular graph and the maximum degree in one part is ? and the maximum degree in the other part is ?.
In the paper we study the problem of interval edge colorings of (k*,2*)-bipartite graphs, for k ? {3,4,5}, and of (5*,3*)-bipartite graphs. We prove that every (5*,2*)-bipartite graph admits an interval edge coloring using at most 6 colors, which can be found in O(n^{3/2}) time, and we prove that an interval edge 5-coloring of a (5*,2*)-bipartite graph can be found in O(n^{3/2}) time, if it exists. We show that every (4^*,2^*)-bipartite graph admits an interval edge 4-coloring, which can be found in O(n) time. The two following problems of interval edge coloring are known to be NP-complete: 6-coloring of (6,3)-biregular graphs (Asratian and Casselgren (2006)) and 5-coloring of (5*,5*)-bipartite graphs (Giaro (1997)). In the paper we prove NP-completeness of 5-coloring of (5*,3*)-bipartite graphs
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
INTERVAL EDGE-COLORING OF COMPLETE AND COMPLETE BIPARTITE GRAPHS WITH RESTRICTIONS
An edge-coloring of a graph G with consecutive integers c1,…,ct is called an interval t-coloring, if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. In this paper, we consider the case where there are restrictions on the edges, and the edge-coloring should satisfy these restrictions. We show that the problem is NP-complete for complete and complete bipartite graphs. We also provide a polynomial solution for a subclass of complete bipartite graphs when the restrictions are on the vertices.An edge-coloring of a graph G with consecutive integers c1,…,ct is called an interval t-coloring, if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. In this paper, we consider the case where there are restrictions on the edges, and the edge-coloring should satisfy these restrictions. We show that the problem is NP-complete for complete and complete bipartite graphs. We also provide a polynomial solution for a subclass of complete bipartite graphs when the restrictions are on the vertices
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