PDF

Discussiones Mathematicae Graph Theory 31(2) (2011) 283-292
DOI: https://doi.org/10.7151/dmgt.1545

Monochromatic Cycles and Monochromatic Paths in arc-colored Digraphs

Hortensia Galeana-Sánchez Guadalupe Gaytán-Gómez Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria, México, D.F. 04510, México

Roc'io Rojas-Monroy Facultad de Ciencias Universidad Autónoma del Estado de México Instituto Literario No. 100, Centro 50000, Toluca, Edo. de México, México

Abstract

We call the digraph D an m-colored digraph if the arcs of D are colored with m colors. A path (or a cycle) is called monochromatic if all of its arcs are colored alike. A cycle is called a quasi-monochromatic cycle if with at most one exception all of its arcs are colored alike. A subdigraph H in D is called rainbow if all its arcs have different colors. 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. The closure of D, denoted by ℭ(D), is the m-colored multidigraph defined as follows: V(ℭ(D)) = V(D), A(ℭ(D)) = A(D)∪{(u,v) with color i |  there exists a uv-monochromatic path colored i contained in D}. Notice that for any digraph D, ℭ (ℭ(D)) ≅ ℭ(D) and D has a kernel by monochromatic paths if and only if ℭ(D) has a kernel.

Let D be a finite m-colored digraph. Suppose that there is a partition C = C₁∪C₂ of the set of colors of D such that every cycle in the subdigraph D[C_i] spanned by the arcs with colors in C_i is monochromatic. We show that if ℭ(D) does not contain neither rainbow triangles nor rainbow P₃ involving colors of both C₁ and C₂, then D has a kernel by monochromatic paths.

This result is a wide extension of the original result by Sands, Sauer and Woodrow that asserts: Every 2-colored digraph has a kernel by monochromatic paths (since in this case there are no rainbow triangles in ℭ(D)).

Keywords: kernel, kernel by monochromatic paths, monochromatic cycles

2010 Mathematics Subject Classification: 05C20.

References

[1]	C. Berge, Graphs (North-Holland, Amsterdam, 1985).
[2]	P. Duchet, Graphes Noyau -- Parfaits, Ann. Discrete Math. 9 (1980) 93--101, doi: 10.1016/S0167-5060(08)70041-4.
[3]	P. Duchet, Classical Perfect Graphs, An introduction with emphasis on triangulated and interval graphs, Ann. Discrete Math. 21 (1984) 67--96.
[4]	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.
[5]	H. Galeana-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.
[6]	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.
[7]	H. Galeana-Sánchez, On monochromatic paths and monochromatics 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. Galeana-Sánchez and J.J. Garcia-Ruvalcaba, Kernels in the closure of coloured digraphs, Discuss. Math. Graph Theory 20 (2000) 243--254 , doi: 10.7151/dmgt.1123.
[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]	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.
[12]	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.
[13]	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.

Received 26 November 2009
>Revised 18 December 2010
Accepted 19 December 2010