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Intrinsic linking in directed graphs. (English) Zbl 1337.57016

This work extends the notion of intrinsic linking to directed graphs. A directed graph \(G\) is called intrinsically linked if it contains a nontrivial link consisting of a pair of consistently oriented directed cycles in every spatial embedding. The double directed version of a graph \(G\), denoted \(\overline{D(G)}\), is obtained by replacing each edge \(e\) of \(G\) by a pair of directed edges with opposite orientations joining the same pair of vertices as \(e\). The main result of this paper is that the double directed version of a graph \(G\) is intrinsically linked if and only if \(G\) is intrinsically linked.
As a corollary to this result it is shown that the complete symmetric directed graph on six vertices, \(\overline {J_6}\), is intrinsically linked, thus extending the work of Conway, Gordon and Sachs [J. H. Conway and C. McA. Gordon, J. Graph Theory 7, 445–453 (1983; Zbl 0524.05028)], [H. Sachs, in: Finite and infinite sets, 6th Hung. Combin. Colloq., Eger/Hung. 1981, Vol. II, Colloq. Math. Soc. János Bolyai 37, 649–662 (1984; Zbl 0568.05026)]. Several other results are shown for \(\overline {J_6}\) including the number of allowable edges that can be deleted to make the subgraph intrinsically linked or not.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
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References:

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