# Kernels by monochromatic paths and the color-class digraph

Discussiones Mathematicae Graph Theory (2011)

- Volume: 31, Issue: 2, page 273-281
- ISSN: 2083-5892

## Access Full Article

top## Abstract

top## How to cite

topHortensia Galeana-Sánchez. "Kernels by monochromatic paths and the color-class digraph." Discussiones Mathematicae Graph Theory 31.2 (2011): 273-281. <http://eudml.org/doc/271051>.

@article{HortensiaGaleana2011,

abstract = {
An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored 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 ≠ 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-colored 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-colored digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph.
},

author = {Hortensia Galeana-Sánchez},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {kernel; kernel by monochromatic paths; the color-class digraph; monochromatic paths; color-class digraph},

language = {eng},

number = {2},

pages = {273-281},

title = {Kernels by monochromatic paths and the color-class digraph},

url = {http://eudml.org/doc/271051},

volume = {31},

year = {2011},

}

TY - JOUR

AU - Hortensia Galeana-Sánchez

TI - Kernels by monochromatic paths and the color-class digraph

JO - Discussiones Mathematicae Graph Theory

PY - 2011

VL - 31

IS - 2

SP - 273

EP - 281

AB -
An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored 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 ≠ 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-colored 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-colored digraph has a kernel by monochromatic paths, in case that the digraph D be a strongly connected digraph.

LA - eng

KW - kernel; kernel by monochromatic paths; the color-class digraph; monochromatic paths; color-class digraph

UR - http://eudml.org/doc/271051

ER -

## References

top- [1] J.M. Le Bars, Counterexample of the 0-1 law for fragments of existential second-order logic; an overview, Bull. Symbolic Logic 9 (2000) 67-82, doi: 10.2307/421076. Zbl0958.03022
- [2] J.M. Le Bars, The 0-1 law fails for frame satisfiability of propositional model logic, Proceedings of the 17th Symposium on Logic in Computer Science (2002) 225-234, doi: 10.1109/LICS.2002.1029831.
- [3] C. Berge, Graphs (North-Holland, Amsterdam, 1985).
- [4] E. Boros and V. Gurvich, Perfect graphs, kernels and cores of cooperative games, Discrete Math. 306 (2006) 2336-2354, doi: 10.1016/j.disc.2005.12.031. Zbl1103.05034
- [5] A.S. Fraenkel, Combinatorial game theory foundations applied to digraph kernels, Electronic J. Combin. 4 (2) (1997) #R10. Zbl0884.05045
- [6] A.S. Fraenkel, Combinatorial games: selected bibliography with a succint gourmet introduction, Electronic J. Combin. 14 (2007) #DS2.
- [7] G. Hahn, P. Ille and R. Woodrow, Absorbing sets in arc-coloured tournaments, Discrete Math. 283 (2004) 93-99, doi: 10.1016/j.disc.2003.10.024. Zbl1042.05049
- [8] H. Galeana-Sánchez, On monochromatic paths and monochromatic cycles in edge coloured tournaments, Discrete Math. 156 (1996) 103-112, doi: 10.1016/0012-365X(95)00036-V. Zbl0857.05054
- [9] H. Galeana-Sánchez, Kernels in edge-coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3. Zbl0958.05061
- [10] H. Galeana-Sánchez and R. Rojas-Monroy, A counterexample to a conjecture on edge-coloured tournaments, Discrete Math. 282 (2004) 275-276, doi: 10.1016/j.disc.2003.11.015. Zbl1042.05039
- [11] H. Galeana-Sánchez and R. Rojas-Monroy, On monochromatic paths and monochromatic 4-cycles in edge coloured bipartite tournaments, Discrete Math. 285 (2004) 313-318, doi: 10.1016/j.disc.2004.03.005. Zbl1049.05042
- [12] G. Gutin and J. Bang-Jensen, Digraphs: Theory, Algorithms and Applications (Springer-Verlag, London, 2001). Zbl0958.05002
- [13] T.W. Haynes, T. Hedetniemi and P.J. Slater, Domination in Graphs (Advanced Topics, Marcel Dekker Inc., 1998). Zbl0883.00011
- [14] T.W. Haynes, T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker Inc., 1998). Zbl0890.05002
- [15] J. von Leeuwen, Having a Grundy Numbering is NP-complete, Report 207 Computer Science Department, University Park, PA, 1976, Pennsylvania State University.
- [16] B. Sands, N. Sauer and R. Woodrow, On monochromatic paths in edge-coloured digraphs, J. Combin. Theory (B) 33 (1982) 271-275, doi: 10.1016/0095-8956(82)90047-8. Zbl0488.05036
- [17] I. Włoch, On imp-sets and kernels by monochromatic paths in duplication, Ars Combin. 83 (2007) 93-99. Zbl1174.05114
- [18] I. Włoch, On kernels by monochromatic paths in the corona of digraphs, Cent. Eur. J. Math. 6 (2008) 537-542. Zbl1152.05033

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.