Discussiones Mathematicae Graph Theory 31(2) (2011)
283-292

doi: 10.7151/dmgt.1545

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 |

Let D be a finite m-colored digraph. Suppose that there is a partition
C = C_{1}∪C_{2} 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 P_{3} involving colors of both C_{1} and C_{2}, 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.

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Received 26 November 2009

>Revised 18 December 2010

Accepted 19 December 2010