Discussiones Mathematicae Graph Theory 30(4) (2010) 545-553
doi: 10.7151/dmgt.1512

## MONOCHROMATIC PATHS AND MONOCHROMATIC SETS OF ARCS IN QUASI-TRANSITIVE DIGRAPHS

Hortensia Galeana-Sánchez1, R. Rojas-Monroy2  and  B. Zavala1

1Instituto de Matemáticas
México

Instituto Literario, Centro 50000, Toluca, Edo. de México
México

## Abstract

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. We call the digraph D an m-coloured digraph if each arc of D is coloured by an element of {1,2,...,m} where m ≥ 1. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if there is no monochromatic path between two vertices of N and if for every vertex v not in N there is a monochromatic path from v to some vertex in N. A digraph D is called a quasi-transitive digraph if (u,v) ∈ A(D) and (v,w) ∈ A(D) implies (u,w) ∈ A(D) or (w,u) ∈ A(D). We prove that if D is an m-coloured quasi-transitive digraph such that for every vertex u of D the set of arcs that have u as initial end point is monochromatic and D contains no C3 (the 3-coloured directed cycle of length 3), then D has a kernel by monochromatic paths.

Keywords: m-coloured quasi-transitive digraph, kernel by monochromatic paths.

2010 Mathematics Subject Classification: 05C15, 05C20.

## References

 [1] J. Bang-Jensen and J. Huang, Quasi-transitive digraphs, J. Graph Theory 20 (1995) 141-161, doi: 10.1002/jgt.3190200205. [2] J. Bang-Jensen and J. Huang, Kings in quasi-transitive digraphs, Discrete Math. 185 (1998) 19-27, doi: 10.1016/S0012-365X(97)00179-9. [3] C. Berge, Graphs (North Holland, Amsterdam, New York, 1985). [4] P. Duchet, Graphes noyau-parfaits, Ann. Discrete Math. 9 (1980) 93-101, doi: 10.1016/S0167-5060(08)70041-4. [5] P. Duchet, Classical Perfect Graphs, An introduction with emphasis on triangulated and interval graphs, Ann. Discrete Math. 21 (1984) 67-96. [6] P. Duchet and H. Meyniel, A note on kernel-critical graphs, Discrete Math. 33 (1981) 103-105, doi: 10.1016/0012-365X(81)90264-8. [7] 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. [8] H. Galeana-Sánchez, Kernels in edge coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3. [9] H. Galena-Sánchez and V. Neumann-Lara, On kernels and semikernels of digraphs, Discrete Math. 48 (1984) 67-76, doi: 10.1016/0012-365X(84)90131-6. [10] H. Galeana-Sánchez and V. Neumann-Lara, On kernel-perfect critical digraphs, Discrete Math. 59 (1986) 257-265, doi: 10.1016/0012-365X(86)90172-X. [11] H. Galeana-Sánchez and R. Rojas-Monroy, Kernels in quasi-transitive digraphs, Discrete Math. 306 (2006) 1969-1974, doi: 10.1016/j.disc.2006.02.015. [12] T. Gallai, Transitive orienterbare graphen, Acta Math. Sci. Hung. 18 (1967) 25-66, doi: 10.1007/BF02020961. [13] Ghouilá-Houri, Caractrisation des graphes non orients dont on peut orienter les arretes de maniere a obtenier le graphe d'un relation d'ordre, C.R. Acad. Sci. Paris 254 (1962) 1370-1371. [14] D. Kelly, Comparability graphs, in graphs and order, (ed. I. Rival), Nato ASI Series C. Vol. 147, D. Reidel (1985) 3-40. [15] M. Kucharska, On (k,l)-kernels of orientations of special graphs, Ars Combin. 60 (2001) 137-147. [16] M. Kucharska and M.Kwaśnik, On (k,l)-kernels of superdigraphs of Pm and Cm, Discuss. Math. Graph Theory 21 (2001) 95-109, doi: 10.7151/dmgt.1135. [17] M.Kwaśnik, The generalization of Richardson's Theorem, Discuss. Math. IV (1981) 11-14. [18] M.Kwaśnik, On (k,l)-kernels of exclusive disjunction, Cartesian sum and normal product of two directed graphs, Discuss. Math. V (1982) 29-34. [19] M. Richardson, Solutions of irreflexive relations, Ann. Math. 58 (1953) 573, doi: 10.2307/1969755. [20] M. Richardson, Extensions theorems for solutions of irreflexive relations, Proc. Nat. Acad. Sci. USA 39 (1953) 649, doi: 10.1073/pnas.39.7.649. [21] 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. [22] A. Włoch and I. Włoch, On (k,l)-kernels in generalized products, Discrete Math. 164 (1997) 295-301, doi: 10.1016/S0012-365X(96)00064-7. [23] I. Włoch, On imp-sets and kernels by monochromatic paths in duplication, Ars Combin. 83 (2007) 93-99. [24] 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.