9,247 research outputs found
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
Some results on the palette index of graphs
Given a proper edge coloring of a graph , we define the palette
of a vertex as the set of all colors appearing
on edges incident with . The palette index of is the
minimum number of distinct palettes occurring in a proper edge coloring of .
In this paper we give various upper and lower bounds on the palette index of
in terms of the vertex degrees of , particularly for the case when
is a bipartite graph with small vertex degrees. Some of our results concern
-biregular graphs; that is, bipartite graphs where all vertices in one
part have degree and all vertices in the other part have degree . We
conjecture that if is -biregular, then , and we prove that this conjecture holds for several families of
-biregular graphs. Additionally, we characterize the graphs whose
palette index equals the number of vertices
The bipartite Ramsey numbers
For the given bipartite graphs , the multicolor bipartite
Ramsey number is the smallest positive integer
such that any -edge-coloring of contains a monochromatic subgraph
isomorphic to , colored with the th color for some . We
compute the exact values of the bipartite Ramsey numbers for
On b-colorings and b-continuity of graphs
A b-coloring of G is a proper vertex coloring such that there is a vertex in each color class, which is adjacent to at least one vertex in every other color class. Such a vertex is called a color-dominating vertex. The b-chromatic number of G is the largest k such that there is a b-coloring of G by k colors.
Moreover, if for every integer k, between chromatic number and b-chromatic number, there exists a b-coloring of G by k colors, then G is b-continuous. Determining the b-chromatic number of a graph G and the decision whether the given graph G is b-continuous or not is NP-hard. Therefore, it is interesting to find new results on b-colorings and b-continuity for special graphs.
In this thesis, for several graph classes some exact values as well as bounds of the b-chromatic number were ascertained. Among all we considered graphs whose independence number, clique number, or minimum degree is close to its order as well as bipartite graphs. The investigation of bipartite graphs was based on considering of the so-called bicomplement which is used to determine the b-chromatic number of special bipartite graphs, in particular those whose bicomplement has a simple structure. Then we studied some graphs whose b-chromatic number is close to its t-degree.
At last, the b-continuity of some graphs is studied, for example, for graphs whose b-chromatic number was already established in this thesis. In particular, we could prove that Halin graphs are b-continuous.:Contents
1 Introduction
2 Preliminaries
2.1 Basic terminology
2.2 Colorings of graphs
2.2.1 Vertex colorings
2.2.2 a-colorings
3 b-colorings
3.1 General bounds on the b-chromatic number
3.2 Exact values of the b-chromatic number for special graphs
3.2.1 Graphs with maximum degree at most 2
3.2.2 Graphs with independence number close to its order
3.2.3 Graphs with minimum degree close to its order
3.2.4 Graphs G with independence number plus clique number at most number of vertices
3.2.5 Further known results for special graphs
3.3 Bipartite graphs
3.3.1 General bounds on the b-chromatic number for bipartite graphs
3.3.2 The bicomplement
3.3.3 Bicomplements with simple structure
3.4 Graphs with b-chromatic number close to its t-degree
3.4.1 Regular graphs
3.4.2 Trees and Cacti
3.4.3 Halin graphs
4 b-continuity
4.1 b-spectrum of special graphs
4.2 b-continuous graph classes
4.2.1 Known b-continuous graph classes
4.2.2 Halin graphs
4.3 Further graph properties concerning b-colorings
4.3.1 b-monotonicity
4.3.2 b-perfectness
5 Conclusion
Bibliograph
Coloring sums of extensions of certain graphs
We recall that the minimum number of colors that allow a proper coloring of graph is called the chromatic number of and denoted . Motivated by the introduction of the concept of the -chromatic sum of a graph the concept of -chromatic sum and -chromatic sum are introduced in this paper. The extended graph of a graph was recently introduced for certain regular graphs. This paper furthers the concepts of -chromatic sum and -chromatic sum to extended paths and cycles. Bipartite graphs also receive some attention. The paper concludes with patterned structured graphs. These last said graphs are typically found in chemical and biological structures
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