58 research outputs found
k-colored kernels
We study -colored kernels in -colored digraphs. An -colored digraph
has -colored kernel if there exists a subset of its vertices such
that
(i) from every vertex there exists an at most -colored
directed path from to a vertex of and
(ii) for every there does not exist an at most -colored
directed path between them.
In this paper, we prove that for every integer there exists a -colored digraph without -colored kernel and if every directed
cycle of an -colored digraph is monochromatic, then it has a -colored
kernel for every positive integer We obtain the following results for some
generalizations of tournaments:
(i) -colored quasi-transitive and 3-quasi-transitive digraphs have a %
-colored kernel for every and respectively (we conjecture
that every -colored -quasi-transitive digraph has a % -colored kernel
for every , and
(ii) -colored locally in-tournament (out-tournament, respectively)
digraphs have a -colored kernel provided that every arc belongs to a
directed cycle and every directed cycle is at most -colored
Tournament Directed Graphs
Paired comparison is the process of comparing objects two at a time. A tournament in Graph Theory is a representation of such paired comparison data. Formally, an n-tournament is an oriented complete graph on n vertices; that is, it is the representation of a paired comparison, where the winner of the comparison between objects x and y (x and y are called vertices) is depicted with an arrow or arc from the winner to the other.
In this thesis, we shall prove several results on tournaments. In Chapter 2, we will prove that the maximum number of vertices that can beat exactly m other vertices in an n-tournament is min{2m + 1,2n - 2m - 1}. The remainder of this thesis will deal with tournaments whose arcs have been colored. In Chapter 3 we will define what it means for a k-coloring of a tournament to be k-primitive. We will prove that the maximum k such that some strong n-tournament can be k-colored to be k-primitive lies in the interval [(n-12), (n2) - [n/4]). In Chapter 4, we shall prove special cases of the following 1982 conjecture of Sands, Sauer, and Woodrow from [14]: Let T be a 3-arc-colored tournament containing no 3-cycle C such that each arc in C is a different color. Then T contains a vertex v such that for any other vertex x, x has a monochromatic path to v
07281 Abstracts Collection -- Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs
From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
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