# Monochromatic cycles and monochromatic paths in arc-colored digraphs

Hortensia Galeana-Sánchez; Guadalupe Gaytán-Gómez; Rocío Rojas-Monroy

Discussiones Mathematicae Graph Theory (2011)

- Volume: 31, Issue: 2, page 283-292
- ISSN: 2083-5892

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topHortensia Galeana-Sánchez, Guadalupe Gaytán-Gómez, and Rocío Rojas-Monroy. "Monochromatic cycles and monochromatic paths in arc-colored digraphs." Discussiones Mathematicae Graph Theory 31.2 (2011): 283-292. <http://eudml.org/doc/270813>.

@article{HortensiaGaleana2011,

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)).},

author = {Hortensia Galeana-Sánchez, Guadalupe Gaytán-Gómez, Rocío Rojas-Monroy},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {kernel; kernel by monochromatic paths; monochromatic cycles; monochromatic paths},

language = {eng},

number = {2},

pages = {283-292},

title = {Monochromatic cycles and monochromatic paths in arc-colored digraphs},

url = {http://eudml.org/doc/270813},

volume = {31},

year = {2011},

}

TY - JOUR

AU - Hortensia Galeana-Sánchez

AU - Guadalupe Gaytán-Gómez

AU - Rocío Rojas-Monroy

TI - Monochromatic cycles and monochromatic paths in arc-colored digraphs

JO - Discussiones Mathematicae Graph Theory

PY - 2011

VL - 31

IS - 2

SP - 283

EP - 292

AB - 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)).

LA - eng

KW - kernel; kernel by monochromatic paths; monochromatic cycles; monochromatic paths

UR - http://eudml.org/doc/270813

ER -

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