H-paths in 2-colored tournaments

Discrete Applied Mathematics, 2010

A digraph D is a union of quasi-transitive digraphs if its arcs can be partitioned into sets A 1 and A 2 such that the induced subdigraph D[A i ] (i = 1, 2) is quasi-transitive. Let D be an m-colored asymmetric union of quasi-transitive digraphs such that every chromatic class is completely included in D[A i ] for some i = 1, 2 and is quasi-transitive. We show that if D does not contain 3-colored triangles (directed cycles and transitive subtournaments of order 3), then D has a kernel by monochromatic directed paths.

AKCE International Journal of Graphs and Combinatorics, 2015

A digraph D is an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), then colour (a) will denote the colour has been used on a. A path (or a cycle) is monochromatic if all of its arcs are coloured alike. A set N ⊆ V (D) is 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 path between them and; (ii) for every vertex x ∈ V (D) \ N there is a vertex y ∈ N such that there is an x y-monochromatic path. The closure of D, denoted by C (D), is the m-coloured multidigraph defined as follows: V (C (D)) = V (D), A(C (D)) = A(D) ∪ {(u, v) with colour i| there is an uv-path coloured i contained in D}. A subdigraph H in D is rainbow if all of its arcs have different colours. A digraph D is transitive by monochromatic paths if the existence of an x y-monochromatic path and an yz-monochromatic path in D imply that there is an x z-monochromatic path in D. We will denote by − → P 3 the path of length 3 and by − → C 3 the cycle of length 3. Let D be a finite m-coloured digraph. Suppose that C is the set of colours used in A(D), and ζ = {C 1 , C 2 ,. .. , C k } (k ≥ 2) is a partition of C, such that for every i ∈ {1, 2,. .. , k} happens that H i = D[{a ∈ A(D) | colour (a) ∈ C i }] is transitive by monochromatic paths. Let {ζ 1 , ζ 2 } be a partition of ζ , and D i will be the spanning subdigraph of D such that A(D i) = {a ∈ A(D) | colour (a) ∈ C j f or some C j ∈ ζ i }. In this paper, we give some sufficient conditions for the existence of a kernel by monochromatic paths in a digraph with the structure mentioned above. In particular we obtain extensions of the following two original results: The result by Sands et al. (1982) that asserts: Every 2-coloured digraph has a kernel by monochromatic paths, and the result by Galeana-Sánchez et al. (2011) that asserts: If D is a finite m-coloured digraph that admits a partition {C 1 , C 2 } of the set of colours of D such that for each i ∈ {1, 2} every cycle in the subdigraph D[C i ] spanned by the arcs with colours in C i is monochromatic, C (D) does not contain neither rainbow triangles nor rainbow − → P 3 (path of length 3) involving colours of both C 1 and C 2 ; then D has a kernel by monochromatic paths. c

Transactions on Combinatorics, 2020

Let $H$ be a digraph possibly with loops and $D$ a digraph without loops whose arcs are colored with the vertices of $H$ ($D$ is said to be an $H$-colored digraph)‎. ‎A directed walk $W$ in $D$ is said to be an $H$-walk if and only if the consecutive colors encountered on $W$ form a directed walk in $H$‎. ‎A subset $N$ of the vertices of $D$ is said to be an $H$-kernel by walks if (1) for every pair of different vertices in $N$ there is no $H$-walk between them ($N$ is $H$-independent by walks) and (2) for each vertex $u$ in $V$($D$)-$N$ there exists an $H$-walk from $u$ to $N$ in $D$ ($N$ is $H$-absorbent by walks)‎. ‎Suppose that $D$ is a digraph possibly infinite‎. ‎In this paper we will work with the subdivision digraph $S_H$($D$) of $D$‎, ‎where $S_H$($D$) is an $H$-colored digraph defined as follows‎: ‎$V$($S_H$($D$)) = $V$($D$) $cup$ $A$($D$) and $A$($S_H$($D$)) = {($u$,$a$)‎ : ‎$a$ = ($u$,$v$) $in$ $A$($D$)} $cup$ {($a$,$v$)‎ : ‎$a$ = ($u$,$v$) $in$ $A$($D$)}‎, ‎where ($u$‎,...

Discrete Mathematics, 1996

The least number of colors needed to color the vertices of a graph G such that the vertices in each color class induces a linear forest is called the path-chromatic number of G, denoted by Zoo (G). If all such colorings of the vertices of G induce the same partitioning of the vertices of G, we say that G is path-chromatically unique. We prove here that there exist infinitely many path-chromatically unique graphs with path-chromatic number t, for each t >/1. We also show that pathchromatically unique graphs of order n and path-chromatic number t >i 2 exist only for n 1> 4t-1.

Discussiones Mathematicae Graph Theory, 2005

We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. Let D be an m-coloured digraph. A set N ⊆ 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 each vertex x ∈ (V (D) − N) there is a vertex y ∈ N such that there is an xy-monochromatic directed path. In this paper is defined the monochromatic path digraph of D, MP (D), and the inner m-colouration of MP (D). Also it is proved that if D is an m-coloured digraph without monochromatic directed cycles, then the number of kernels by monochromatic paths in D is equal to the 408

Lecture Notes in Computer Science, 2009

We provide a new proof of a theorem of Saks which is an extension of Greene's Theorem to acyclic digraphs, by reducing it to a similar, known extension of Greene and Kleitman's Theorem. This suggests that the Greene-Kleitman Theorem is stronger than Greene's Theorem on posets. We leave it as an open question whether the same holds for all digraphs, that is, does Berge's conjecture concerning path partitions in digraphs imply the extension of Greene's theorem to all digraphs (conjectured by Aharoni, Hartman and Hoffman)?

Discrete Applied Mathematics

We study $k$-colored kernels in $m$-colored digraphs. An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K$ of its vertices such that (i) from every vertex $v\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and (ii) for every $u,v\in K$ there does not exist an at most $k$-colored directed path between them. In this paper, we prove that for every integer $k\geq 2$ there exists a $% (k+1)$-colored digraph $D$ without $k$-colored kernel and if every directed cycle of an $m$-colored digraph is monochromatic, then it has a $k$-colored kernel for every positive integer $k.$ We obtain the following results for some generalizations of tournaments: (i) $m$-colored quasi-transitive and 3-quasi-transitive digraphs have a $k$% -colored kernel for every $k\geq 3$ and $k\geq 4,$ respectively (we conjecture that every $m$-colored $l$-quasi-transitive digraph has a $k$% -colored kernel for every $k\geq l+1)$, and (ii) $m$-colored local...

Pacific Journal of Mathematics, 1985

Let G be an acyclic directed graph with \V(G)\>k. We prove that there exists a colouring { C x , C 2 ,..., C m } such that for every collection {P l9 P 2 ,... ,P k } of k vertex disjoint paths with |UjLi Pj\ a maximum, each colour class C, meets min{|CJ, k} of these paths. An analogous theorem, partially interchanging the roles of paths and colour classes, has been shown by Cameron [4] and Saks [17] and we indicate a third proof.

Discussiones Mathematicae Graph Theory, 2010

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. We call the digraph D an m-coloured digraph if each arc of D is coloured by an element of {1, 2,. .. , m} where m ≥ 1. 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 there is no monochromatic path between two vertices of N and if for every vertex v not in N there is a monochromatic path from v to some vertex in N. A digraph D is called a quasi-transitive digraph if (u, v) ∈ A(D) and (v, w) ∈ A(D) implies (u, w) ∈ A(D) or (w, u) ∈ A(D). We prove that if D is an m-coloured quasi-transitive digraph such that for every vertex u of D the set of arcs that have u as initial end point is monochromatic and D contains no C 3 (the 3-coloured directed cycle of length 3), then D has a kernel by monochromatic paths.

Discrete Mathematics, 2008

Let D be an edge-coloured digraph, V (D) will denote the set of vertices of D; a set N ⊆ V (D) is said to be a kernel by monochromatic paths of D if it satisfies the following two conditions: For every pair of different vertices u, v ∈ N there is no monochromatic directed path between them and; for every vertex x ∈ V (D) − N there is a vertex y ∈ N such that there is an x y-monochromatic directed path. In this paper we consider some operations on edge-coloured digraphs, and some sufficient conditions for the existence or uniqueness of kernels by monochromatic paths of edge-coloured digraphs formed by these operations from another edge-coloured digraphs.