Discussiones Mathematicae Graph Theory 33(3) (2013)
493-507
DOI: https://doi.org/10.7151/dmgt.1695
γ-cycles and transitivity by monochromatic paths in arc-coloured digraphs
Enrique Casas-Bautista(1) Hortensia Galeana-Sánchez(2) and Rocío Rojas-Monroy(1)
(1) Facultad de Ciencias, Universidad Autónoma del Estado de México |
Abstract
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.
Keywords: digraph, kernel, kernel by monochromatic paths, γ-cycle
2010 Mathematics Subject Classification: 05C20, 05C38, 05C69.
References
[1] | J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications (Springer, London, 2001). |
[2] | C. Berge, Graphs (North-Holland, Amsterdam, 1985). |
[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. |
[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. |
[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. |
[6] | P. Duchet, Graphes noyau - parfaits, Ann. Discrete Math. 9 (1980) 93--101, doi: 10.1016/S0167-5060(08)70041-4. |
[7] | P. Duchet, Classical perfect graphs, An introduction with emphasis on triangulated and interval graphs, Ann. Discrete Math. 21 (1984) 67--96. |
[8] | P. Duchet and H. Meynel, A note on kernel-critical graphs, Discrete Math. 33 (1981) 103--105, doi: 10.1016/0012-365X(81)90264-8. |
[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. |
[10] | H. Galena-Sánchez, Kernels in edge-coloured digraphs, Discrete Math. 184 (1998) 87--99, doi: 10.1016/S0012-365X(97)00162-3. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] | J.M. Le Bars, Counterexample of the 0-1 law for fragments of existential second-order logic; an overview, Bull. Symbolic Logic 6 (2000) 67--82, doi: 10.2307/421076. |
[24] | J.M. Le Bars, The 0-1 law fails for frame satisfiability of propositional model logic, in: Proceedings of the 17th Symposium on Logic in Computer Science (2002) 225--234, doi: 10.1109/LICS.2002.1029831. |
[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. |
[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. |
[29] | I. Włoch, On kernels by monochromatic paths in the corona of digraphs, Cent. Eur. J. Math. 6 (2008) 537--542, doi: 10.2478/s11533-008-0044-6. |
[30] | I. Włoch, On imp-sets and kernels by monochromatic paths in duplication, Ars Combin. 83 (2007) 93--99. |
Received 31 January 2012
Revised 15 April 2013
Accepted 15 April 2013
Close