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2013 | 33 | 3 | 493-507
Tytuł artykułu

γ-Cycles And Transitivity By Monochromatic Paths In Arc-Coloured Digraphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . . , un), such that ui ≠ uj if i ≠ j and for every i ∈ {0, 1, . . . , n} there is a uiui+1-monochromatic path in D and there is no ui+1ui-monochromatic path in D (the indices of the vertices will be taken mod n+1). 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 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 path. Let D be a finite m-coloured digraph. Suppose that {C1,C2} is a partition of C, the set of colours of D, and Di will be the spanning subdigraph of D such that A(Di) = {a ∈ A(D) | colour(a) ∈ Ci}. 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 an extension of the original result by B. Sands, N. Sauer and R. Woodrow that asserts: Every 2-coloured digraph has a kernel by monochromatic paths. Also, we extend other results obtained before where it is proved that under some conditions an m-coloured digraph has no γ-cycles.
Słowa kluczowe
Wydawca
Rocznik
Tom
33
Numer
3
Strony
493-507
Opis fizyczny
Daty
wydano
2013-07-01
online
2013-07-30
Twórcy
  • Facultad de Ciencias, Universidad Autónoma del Estado de México Instituto Literario No. 100, Centro, 50000 Toluca, Edo. de México, México, ecasasb@uaemex.mx
  • Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria, México, D.F. 04510, México, hgaleana@matem.unam.mx
  • Facultad de Ciencias, Universidad Autónoma del Estado de México Instituto Literario No. 100, Centro, 50000 Toluca, Edo. de México, México, mrrm@uaemex.mx
Bibliografia
  • [1] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications (Springer, London, 2001).
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  • [3] C. Berge and P. Duchet, Recent problems and results about kernels in directed graphs, Discrete Math. 86 (1990) 27-31. doi:10.1016/0012-365X(90)90346-J[Crossref]
  • [4] P. Delgado-Escalante and H. Galena-Sánchez, Kernels and cycles’ subdivisions in arc-colored tournaments, Discuss. Math. Graph Theory 29 (2009) 101-117. doi:10.7151/dmgt.1435[Crossref]
  • [5] P. Delgado-Escalante and H. Galena-Sánchez, On monochromatic paths and bicolored subdigraphs in arc-colored tournaments, Discuss. Math. Graph Theory 31 (2011) 791-820. doi:10.7151/dmgt.1580[Crossref]
  • [6] P. Duchet, Graphes noyau - parfaits, Ann. Discrete Math. 9 (1980) 93-101. doi:10.1016/S0167-5060(08)70041-4[Crossref]
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  • [9] H. Galena-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[Crossref]
  • [10] H. Galena-Sánchez, Kernels in edge-coloured digraphs, Discrete Math. 184 (1998) 87-99. doi:10.1016/S0012-365X(97)00162-3[WoS][Crossref]
  • [11] H. Galena-Sánchez and J.J. García-Ruvalcaba, Kernels in the closure of coloured digraphs, Discuss. Math. Graph Theory 20 (2000) 243-254. doi:10.7151/dmgt.1123[Crossref]
  • [12] H. Galeana-Sánchez, J.J. García-Ruvalcaba, On graphs all of whose {C3, T3}-free arc colorations are kernel perfect, Discuss. Math. Graph Theory 21 (2001) 77-93. doi:10.7151/dmgt.1134 [Crossref]
  • [13] H. Galena-Sánchez, G. Gaytán-Gómez and R. Rojas-Monroy, Monochromatic cycles and monochromatic paths in arc-coloured digraphs, Discuss. Math. Graph Theory 31 (2011) 283-292. doi:10.7151/dmgt.1545[Crossref]
  • [14] H. Galena-Sánchez, V. Neumann-Lara, On kernels and semikernels of digraphs, Discrete Math. 48 (1984) 67-76. doi:10.1016/0012-365X(84)90131-6[Crossref]
  • [15] H. Galeana-Sánchez, V. Neumann-Lara, On kernel-perfect critical digraphs, Discrete Math. 59 (1986) 257-265. doi:10.1016/0012-365X(86)90172-X[Crossref]
  • [16] 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[Crossref]
  • [17] 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[Crossref]
  • [18] H. Galeana-Sánchez, R. Rojas-Monroy, Independent domination by monochromatic paths in arc coloured bipartite tournaments, AKCE J. Graphs. Combin. 6 (2009) 267-285.
  • [19] H. Galeana-Sánchez and R. Rojas-Monroy, Monochromatic paths and monochromatic cycles in edge-coloured k-partite tournaments, Ars Combin. 97A (2010) 351-365.
  • [20] H. Galena-Sánchez, R. Rojas-Monroy and B. Zavala, Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs, Discuss. Math. Graph Theory 29 (2009) 337-347. doi:10.7151/dmgt.1450
  • [21] H. Galena-Sánchez, R. Rojas-Monroy and B. Zavala, Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs, Discuss. Math. Graph Theory 30 (2010) 545-553. doi:10.7151/dmgt.1512[Crossref]
  • [22] 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[Crossref]
  • [23] J.M. Le Bars, Counterexample of the 0 − 1 law for fragments of existential secondorder logic; an overview, Bull. Symbolic Logic 6 (2000) 67-82. doi:10.2307/421076[Crossref]
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  • [25] J. von Leeuwen, Having a Grundy numbering is NP-complete, Report 207 Computer Science Department, Pennsylvania State University, University Park, PA (1976).
  • [26] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1944).
  • [27] 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[Crossref]
  • [28] S. Minggang, On monochromatic paths in m-coloured tournaments, J. Combin, Theory (B) 45 (1988) 108-111. doi:10.1016/0095-8956(88)90059-7[Crossref]
  • [29] I. Włoch, On kernels by monochromatic paths in the corona of digraphs, Cent. Eur. J. Math. 6 (2008) 537-542. doi:10.7151/s11533-008-0044-6[Crossref]
  • [30] I. Włoch, On imp-sets and kernels by monochromatic paths in duplication, Ars Combin. 83 (2007) 93-99.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1695
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