Kernels in the closure of coloured digraphs

• Autorzy: Hortensia Galeana-Sánchez , José de Jesús García-Ruvalcaba
• Języki publikacji: EN

Abstrakty

EN

Let D be a digraph with V(D) and A(D) the sets of vertices and arcs of D, respectively. A kernel of D is a set I ⊂ V(D) such that no arc of D joins two vertices of I and for each x ∈ V(D)∖I there is a vertex y ∈ I such that (x,y) ∈ A(D). A digraph is kernel-perfect if every non-empty induced subdigraph of D has a kernel. If D is edge coloured, we define the closure ξ(D) of D the multidigraph with V(ξ(D)) = V(D) and $A(ξ(D)) = ⋃_i{(u,v)$ with colour i there exists a monochromatic path of colour i from the vertex u to the vertex v contained in D}.
Let T₃ and C₃ denote the transitive tournament of order 3 and the 3-cycle, respectively, both of whose arcs are coloured with 3 different colours. In this paper, we survey sufficient conditions for the existence of kernels in the closure of edge coloured digraphs, also we prove that if D is obtained from an edge coloured tournament by deleting one arc and D does not contain T₃ or C₃, then ξ(D) is a kernel-perfect digraph.

Słowa kluczowe

EN

kernel; closure; tournament

Kategorie tematyczne

• 05C20: Directed graphs (digraphs), tournaments

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2000
• Tom: 20
• Numer: 2
• Strony: 243-254

Daty

wydano

2000

otrzymano

2000-01-31

poprawiono

2000-10-02

Twórcy

autor

Hortensia Galeana-Sánchez

• Instituto de Matemáticas, UNAM, Circuito Exterior, 04510 México, D.F. México

autor

José de Jesús García-Ruvalcaba

• Facultad de Ciencias, Universidad Autónma de Baja California, Km. 103 Carretera Tijuana-Ensenada, Apdo Postal # 1880, 22830 Ensenada, B.C. México

Bibliografia

• [1] C. Berge and P. Duchet, Recent problems and results about kernels in directed graphs, Discrete Math. 86 (1990) 27-31, doi: 10.1016/0012-365X(90)90346-J.
• [2] 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.
• [3] H. Galeana-Sánchez, Kernels in edge coloured digraphs, Discrete Math. 184 (1998) 87-99, doi: 10.1016/S0012-365X(97)00162-3.
• [4] H. Galeana-Sánchez and J.J. García-Ruvalcaba, On graph all of whose {C₃, T₃}-free arc colorations are kernel-perfect, submitted.
• [5] Shen Minggang, On monochromatic paths in m-coloured tournaments, J. Combin. Theory (B) 45 (1988) 108-111, doi: 10.1016/0095-8956(88)90059-7.
• [6] 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.

Identyfikatory

DOI

10.7151/dmgt.1123