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Kernels in the closure of coloured digraphs

100%
Galeana-Sánchez H., García-Ruvalcaba J.
Discussiones Mathematicae Graph Theory
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2000
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tom 20
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nr 2
243-254
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.
2
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New sufficient conditions for hamiltonian and pancyclic graphs

100%
Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
29-38
EN
For a graph G of order n we consider the unique partition of its vertex set V(G) = A ∪ B with A = {v ∈ V(G): d(v) ≥ n/2} and B = {v ∈ V(G):d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.
3
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Closure for spanning trees and distant area

100%
Fujisawa J., Saito A., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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nr 1
143-159
EN
A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with $deg_G u + deg_G v ≥ n-1$, G has a spanning k-ended tree if and only if G+uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on $deg_G u + deg_G v$ and the structure of the distant area for u and v. We prove that if the distant area contains $K_r$, we can relax the lower bound of $deg_G u + deg_G v$ from n-1 to n-r. And if the distant area itself is a complete graph and G is 2-connected, we can entirely remove the degree sum condition.
4
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The Ryjáček Closure and a Forbidden Subgraph

88%
Saito A., Xiong L.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 3
621-628
EN
The Ryjáček closure is a powerful tool in the study of Hamiltonian properties of claw-free graphs. Because of its usefulness, we may hope to use it in the classes of graphs defined by another forbidden subgraph. In this note, we give a negative answer to this hope, and show that the claw is the only forbidden subgraph that produces non-trivial results on Hamiltonicity by the use of the Ryjáček closure.
5
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On traceability and 2-factors in claw-free graphs

88%
Fronček D., Ryjáček Z., Skupień Z.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
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nr 1
55-71
EN
If G is a claw-free graph of sufficiently large order n, satisfying a degree condition σₖ > n + k² - 4k + 7 (where k is an arbitrary constant), then G has a 2-factor with at most k - 1 components. As a second main result, we present classes of graphs 𝓒₁,...,𝓒₈ such that every sufficiently large connected claw-free graph satisfying degree condition σ₆(k) > n + 19 (or, as a corollary, δ(G) > (n+19)/6) either belongs to $⋃ ⁸_{i=1} 𝓒_i$ or is traceable.
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