Discussiones Mathematicae Graph Theory 31(2) (2011) 273-281
doi: 10.7151/dmgt.1544

[BIBTex] [PDF] [PS]

Kernels by monochromatic paths
and the color-class digraph

Hortensia Galeana-Sánchez

Instituto de Matemáticas
Universidad Nacional Autónoma de México
Area de la Investigación Cientifica
Ciudad Universitaria
04510, México, D.F., México


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:

For any x,y ∈ S, x ≠ y, there is no monochromatic directed path between them.
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:

Every closed directed walk has an even number of color changes,
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.

Keywords: kernel, kernel by monochromatic paths, the color-class digraph

2010 Mathematics Subject Classification: 05C20.


[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.
[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.
[5]A.S. Fraenkel, Combinatorial game theory foundations applied to digraph kernels, Electronic J. Combin. 4 (2) (1997) #R10.
[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.
[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.
[9]H. Galeana-Sánchez, Kernels in edge-coloured digraphs, Discrete Math. 184 (1998) 87--99, doi: 10.1016/S0012-365X(97)00162-3.
[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.
[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.
[12]G. Gutin and J. Bang-Jensen, Digraphs: Theory, Algorithms and Applications (Springer-Verlag, London, 2001).
[13]T.W. Haynes, T. Hedetniemi and P.J. Slater, Domination in Graphs (Advanced Topics, Marcel Dekker Inc., 1998).
[14]T.W. Haynes, T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker Inc., 1998).
[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.
[17]I. Włoch, On imp-sets and kernels by monochromatic paths in duplication, Ars Combin. 83 (2007) 93--99.

Received 24 November 2009
Revised 2 December 2010
Accepted 27 January 2011