Hortensia Galeana-Sánchez

On kernels and semikernels of digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w @? V(D) - N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be an R-digraph. Minimal non-R-digraphs are called R^--digraphs. In this paper some structural results concerning R^--digraphs and sufficient conditions for a digraph to be an R-digraph are presented. In particular, it is proved that every vertex (resp. arc) in an R^--digraph is contained in an odd directed cycle not containing special pseudodiagonals. It is also proved that any digraph in which every odd directed cycle has two pseudodiagonals with consecutive terminal endpoints is an R-digraph. Previous results of other authors (Richardson, Meyniel, Duchet, and others) are generalized.

On kernel-perfect critical digraphs

In this paper we investigate new sufficient conditions for a digraph to be kernel-perfect (KP) and some structural properties of kernel-perfect critical (KPC) digraphs. In particular, it is proved that the asymmetrical part of any KPC digraph is strongly connected. A new method to construct KPC digraphs is developed. The existence of KP and KPC digraphs with arbitrarily large dichromatic number is also discussed.

On monochromatic paths and monochromatic cycles in edge coloured tournaments

Abstract We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. A directed cycle is called a quasi-monochromatic cycle if with at most one exception all of its arcs are coloured alike. In this paper I obtain some sufficient conditions for an m -coloured tournament T to have a vertex v such that for every other vertex x of T there is a monochromatic directed path from x to v . In particular I prove: If T is an m -coloured tournament such that every directed cycle of length at most 4 is a quasi-monochromatic cycle then there is a vertex v of T such that for every other vertex x of T there is a monochromatic directed path from x to v .

Kernels in edge-colored digraphs

Abstract We call the digraph D an m -coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A directed cycle is called quasi-monochromatic if with at most one exception all of its arcs are coloured alike. A set N C V(D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u , v ∈ N there is no monochromatic directed path between them and; (ii) for every vertex x ∈ V ( D )− N there is a vertex y ∈ N such that there is an xy -monochromatic directed path. In this paper I survey sufficient conditions for a m -coloured digraph to have a kernel by monochromatic paths. I also prove that if D is an m -coloured digraph resulting from the deletion of a single arc of some m -coloured tournament and every directed cycle of length at most 4 is quasi-monochromatic then D has a kernel by monochromatic paths.

Monochromatic paths and monochromatic sets of arcs in bipartite tournaments

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours and all of them are used. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices there is no monochromatic path between them and for every vertex v in V (D) \ N there is a monochromatic path from v to some vertex in N . We denote by A(u) the set of arcs of D that have u as the initial endpoint. In this paper we introduce the concept of semikernel modulo i by monochromatic paths of an m-coloured digraph. This concept allow us to find sufficient conditions for the existence of a kernel by monochromatic paths in an m-coloured digraph. In particular we deal with bipartite tournaments such that A(z) is monochromatic for each z ∈ V (D).

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.

Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v / ∈ N there is a monochromatic path from v to N . We denote by A(u) the set of arcs of D that have u as the initial vertex. We prove that if D is an m-coloured 3-quasitransitive digraph such that for every vertex u of D, A(u) is monochromatic and D satisfies some colouring conditions over one subdigraph of D of order 3 and two subdigraphs of D of order 4, then D has a kernel by monochromatic paths.

Kernels by monochromatic paths and the color-class digraph

An m-coloured digraph is a digraph whose arcs are coloured with m colors. A directed path is monochromatic when its arcs are coloured alike. A set S ⊆ V (D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x, y ∈ S, x 6= y, there is no monochromatic directed path between them. 2. For each z ∈ (V (D)− S) there exists a zS-monochromatic directed path In this paper it is introduced the concept of color-class digraph to prove that if D is an m-coloured strongly connected finite digraph such that: (i) Every closed directed walk has an even number of color changes, (ii) Every directed walk starting and ending with the same color has an even number of color changes, then D has a kernel by monochromatic paths. This result generalizes a classical result by Sands, Sauer and Woodrow which asserts that any 2-coloured digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph. 2000 Mathematics Subject classification: 05C20.

Independent sets and non-augmentable paths in generalizations of tournaments

We study different classes of digraphs, which are generalizations of tournaments, to have the property of possessing a maximal independent set intersecting every non-augmentable path (in particular, every longest path). The classes are the arc-local tournament, quasi-transitive, locally in-semicomplete (out-semicomplete), and semicomplete k-partite digraphs. We present results on strongly internally and finally non-augmentable paths as well as a result that relates the degree of vertices and the length of longest paths. A short survey is included in the introduction.

On the existence of kernels and h -kernels in directed graphs

Abstract A directed graph D with vertex set V is called cyclically h -partite ( h ⩾2) provided one can partition V = V 0 + V 1 +⋯+ V h −1 so that if ( u , υ) is an arc of D then uϵV i , and υϵV i +1 (notation mod h ). In this communication we obtain a characterization of cyclically h -partite strongly connected digraphs. As a consequence we obtain a sufficient condition for a digraph to have a h -kernel.