5 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
Independent sets and non-augmentable paths in arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs
AbstractA digraph is arc-locally in-semicomplete if for any pair of adjacent vertices x,y, every in-neighbor of x and every in-neighbor of y either are adjacent or are the same vertex. A digraph is quasi-arc-transitive if for any arc xy, every in-neighbor of x and every out-neighbor of y either are adjacent or are the same vertex. Laborde, Payan and Xuong proposed the following conjecture: Every digraph has an independent set intersecting every non-augmentable path (in particular, every longest path). In this paper, we shall prove that this conjecture is true for arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs
χ-bounded families of oriented graphs
A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets P of orientations of P 4 (the path on four vertices) similar statements hold. We establish some positive and negative results
On the structure of strong 3-quasi-transitive digraphs
In this paper, D = (V (D), A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V (D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u -> v -> w -> z in D, then u and z are adjacent or u = z. In Bang-Jensen (2004)[3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3-quasi-transitive digraphs are the strong semicomplete digraphs and strong semicomplete bipartite digraphs. In this paper, we exhibit a family of strong 3-quasi-transitive digraphs distinct from strong semicomplete digraphs and strong semicomplete bipartite digraphs and provide a complete characterization of strong 3-quasi-transitive digraphs. (C) 2010 Elsevier B.V. All rights reserved