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2011 | 31 | 2 | 273-281
Tytuł artykułu

Kernels by monochromatic paths and the color-class digraph

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
31
Numer
2
Strony
273-281
Opis fizyczny
Daty
wydano
2011
otrzymano
2009-11-24
poprawiono
2010-12-02
zaakceptowano
2011-01-27
Twórcy
  • Instituto de Matemáticas, Universidad Nacional Autónoma de México, Area de la Investigación Cientifica, Ciudad Universitaria, 04510, México, D.F., México
Bibliografia
  • [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.
  • [18] I. Włoch, On kernels by monochromatic paths in the corona of digraphs, Cent. Eur. J. Math. 6 (2008) 537-542.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1544
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