4,434 research outputs found
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
Optimality program in segment and string graphs
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time on string graphs while an algorithm running
in time for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.Comment: 19 pages, 15 figure
List coloring in the absence of a linear forest.
The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The Listk-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u)⊆{1,…,k}. Let Pn denote the path on n vertices, and G+H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that Listk-Coloring can be solved in polynomial time for graphs with no induced rP1+P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1+P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H
Efficient and Perfect domination on circular-arc graphs
Given a graph , a \emph{perfect dominating set} is a subset of
vertices such that each vertex is
dominated by exactly one vertex . An \emph{efficient dominating set}
is a perfect dominating set where is also an independent set. These
problems are usually posed in terms of edges instead of vertices. Both
problems, either for the vertex or edge variant, remains NP-Hard, even when
restricted to certain graphs families. We study both variants of the problems
for the circular-arc graphs, and show efficient algorithms for all of them
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