Tournaments with kernels by monochromatic paths

In this paper we prove the existence of kernels by monochromatic paths in m-coloured tournaments in which every cyclic tournament of order 3 is atmost 2-coloured in addition to other restrictions on the colouring ofcertain subdigraphs. We point out that in all previous results on kernelsby monochromatic paths in arc coloured tournaments, certain smallsubstructures are required to be monochromatic or monochromatic with atmost one exception, whereas here we allow up to three colours in two smallsubstructures.

[1]  Jos'e Luis P'erez Garmendia.  On Weighted Tempered Moving Averages Processes , 2008 .

[2]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[3]  Hortensia Galeana-Sánchez,et al.  Directed hypergraphs: A tool for researching digraphs and hypergraphs , 2009, Discuss. Math. Graph Theory.

[4]  Jean-Marie Le Bars.  Counterexamples of the 0-1 law for fragments of existential second-order logic: an overview , 2000, Bull. Symb. Log..

[5]  Claude Berge,et al.  Recent problems and results about kernels in directed graphs , 1991, Discret. Math..

[6]  H. Galeana-Sánchez,et al.  Independent domination by monochromatic paths in arc coloured bipartite tournaments , 2009 .

[7]  S. E. Markosyan,et al.  ω-Perfect graphs , 1990 .

[8]  Sergio Rajsbaum,et al.  New combinatorial topology bounds for renaming: the lower bound , 2010, Distributed Computing.

[9]  Hortensia Galeana-Sánchez,et al.  H-kernels in the D-join , 2011, Ars Comb..

[10]  Norbert Sauer,et al.  On monochromatic paths in edge-coloured digraphs , 1982, J. Comb. Theory, Ser. B.

[11]  Hortensia Galeana-Sánchez,et al.  A counterexample to a conjecture on edge-coloured tournaments , 2004, Discret. Math..

[12]  Mika Olsen,et al.  Kernels by monochromatic paths in digraphs with covering number 2 , 2011, Discret. Math..

[13]  Hortensia Galeana-Sánchez,et al.  Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments , 2008, Discuss. Math. Graph Theory.

[14]  Hortensia Galeana-Sánchez,et al.  Monochromatic paths and monochromatic sets of arcs in 3-quasitransitive digraphs , 2009, Discuss. Math. Graph Theory.

[15]  Hortensia Galeana-Sánchez,et al.  On monochromatic paths and monochromatic 4-cycles in edge coloured bipartite tournaments , 2004, Discret. Math..

[16]  Barba Flores Luis Felipe,et al.  Shortest H-restricted paths in arc colored digraphs , 2011 .

[17]  Sorin Micu,et al.  A spectral study of the boundary controllability of the linear 2-D wave equation in a rectangle , 2010, Asymptot. Anal..

[18]  Hortensia Galeana-Sánchez,et al.  Monochromatic cycles and monochromatic paths in arc-colored digraphs , 2011, Discuss. Math. Graph Theory.

[19]  Hortensia Galeana-Sánchez.  A new characterization of perfect graphs , 2012, Discret. Math..

[20]  Sergio Rajsbaum,et al.  New combinatorial topology bounds for renaming: The upper bound , 2012, JACM.

[21]  Shen Minggang.  On monochromatic paths in m-coloured tournaments , 1988, J. Comb. Theory, Ser. B.

[22]  Aviezri S. Fraenkel,et al.  Combinatorial Games: selected Bibliography with a Succinct Gourmet Introduction , 2012 .

[23]  Jean-Marie Le Bars,et al.  The 0-1 law fails for frame satisfiability of propositional modal logic , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

[24]  Henry Meyniel,et al.  Kernels in directed graphs: a poison game , 1993, Discret. Math..

[25]  Vladimir Gurvich,et al.  Perfect graphs, kernels, and cores of cooperative games , 2006, Discret. Math..

[26]  Hortensia Galeana-Sánchez,et al.  Monochromatic Paths and at Most 2-Coloured Arc Sets in Edge-Coloured Tournaments , 2005, Graphs Comb..

[27]  Hortensia Galeana-Sánchez,et al.  Monochromatic paths and monochromatic sets of arcs in bipartite tournaments , 2009, Discuss. Math. Graph Theory.

[28]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .

[29]  Hortensia Galeana-Sánchez,et al.  Kernels and some operations in edge-coloured digraphs , 2008, Discret. Math..

[30]  Assia Benabdallah,et al.  The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials , 2011 .

[31]  Camino Balbuena,et al.  A sufficient condition for kernel perfectness of a digraph in terms of semikernels modulo F , 2012 .

[32]  Assia Benabdallah,et al.  Controllability to trajectories for some parabolic systems of three and two equations by one control force , 2013 .

[33]  Amaury Lambert,et al.  Proof(s) of the Lamperti representation of Continuous-State Branching Processes , 2008, 0802.2693.

[34]  S. Hedetniemi,et al.  Domination in graphs : advanced topics , 1998 .

[35]  Mexico.  On the asymptotic behaviour of increasing self-similar Markov processes , 2009 .

[36]  Hortensia Galeana-Sánchez,et al.  Monochromatic paths and monochromatic cycles in edge-coloured k-partite tournaments , 2011 .

[37]  Hortensia Galeana-Sánchez,et al.  A classification of arc-locally semicomplete digraphs , 2009, Electron. Notes Discret. Math..

[38]  Aviezri S. Fraenkel,et al.  Combinatorial game theory foundations applied to digraph kernels , 1996, Electron. J. Comb..

[39]  Hortensia Galeana-S.  On monochromatic paths and monochromatic 4-cycles in edge coloured bipartite tournaments , 2004 .

[40]  L. PASTRANA RAMÍREZ,et al.  INDEPENDENT RESTRICTED DOMINATION AND THE LINE DIGRAPH , 2011 .

[41]  Eli Gafni,et al.  Recursion in Distributed Computing , 2010, SSS.

[42]  Hortensia Galeana-Sánchez,et al.  Monochromatic paths and monochromatic sets of arcs in quasi-transitive digraphs , 2010, Discuss. Math. Graph Theory.

[43]  MICHO DURDEVICH.  GEOMETRY OF QUANTUM PRINCIPAL BUNDLES III Structure of Calculi and Around , 2010 .

[44]  Victor Hernandez-Urbina,et al.  Applying DNA computing to diagnose-and-interfere hepatic fibrosis , 2010, 2010 Sixth International Conference on Natural Computation.

[45]  Hortensia Galeana-Sánchez,et al.  Kernels by monochromatic paths and the color-class digraph , 2011, Discuss. Math. Graph Theory.

[46]  Hortensia Galeana-Sánchez,et al.  On the structure of strong 3-quasi-transitive digraphs , 2010, Discret. Math..