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Independent transversals of longest paths in locally semicomplete and locally transitive digraphs

100%
Galeana-Sánchez H., Gómez R., Montellano-Ballesteros J.
Discussiones Mathematicae Graph Theory
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2009
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tom 29
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nr 3
469-480
EN
We present several results concerning the Laborde-Payan-Xuang conjecture stating that in every digraph there exists an independent set of vertices intersecting every longest path. The digraphs we consider are defined in terms of local semicompleteness and local transitivity. We also look at oriented graphs for which the length of a longest path does not exceed 4.
2
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Independent Detour Transversals in 3-Deficient Digraphs

88%
van Aardt S., Frick M., Singleton J.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 2
261-275
EN
In 1982 Laborde, Payan and Xuong [Independent sets and longest directed paths in digraphs, in: Graphs and other combinatorial topics (Prague, 1982) 173-177 (Teubner-Texte Math., 59 1983)] conjectured that every digraph has an independent detour transversal (IDT), i.e. an independent set which intersects every longest path. Havet [Stable set meeting every longest path, Discrete Math. 289 (2004) 169-173] showed that the conjecture holds for digraphs with independence number two. A digraph is p-deficient if its order is exactly p more than the order of its longest paths. It follows easily from Havet’s result that for p = 1, 2 every p-deficient digraph has an independent detour transversal. This paper explores the existence of independent detour transversals in 3-deficient digraphs.
3
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The directed path partition conjecture

88%
Frick M., van Aardt S., Dlamini G., Dunbar J., Oellermann O.
Discussiones Mathematicae Graph Theory
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2005
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tom 25
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nr 3
331-343
EN
The Directed Path Partition Conjecture is the following: If D is a digraph that contains no path with more than λ vertices then, for every pair (a,b) of positive integers with λ = a+b, there exists a vertex partition (A,B) of D such that no path in D⟨A⟩ has more than a vertices and no path in D⟨B⟩ has more than b vertices. We develop methods for finding the desired partitions for various classes of digraphs.
4
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Detour chromatic numbers

75%
Frick M., Bullock F.
Discussiones Mathematicae Graph Theory
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2001
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tom 21
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nr 2
283-291
EN
The nth detour chromatic number, χₙ(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ( G). We conjecture that χₙ(G) ≤ ⎡(τ(G))/n⎤ for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some sufficient conditions for a graph to have nth chromatic number at most 2.
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