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Spiders everywhere. (English) Zbl 1454.05068

Summary: A spider is a tree with at most one branch (a vertex of degree at least 3) centred at the branch if it exists, and centred at any vertex otherwise. A graph \(G\) is arachnoid if for any vertex \(v\) of \(G\), there exists a spanning spider of \(G\) centred at \(v\) – in other words: there are spiders everywhere! Hypotraceable graphs are non-traceable graphs in which all vertex-deleted subgraphs are traceable. L. Gargano et al. [Discrete Math. 285, No. 1–3, 83–95 (2004; Zbl 1044.05048)] defined arachnoid graphs as natural generalisations of traceable graphs and asked for the existence of arachnoid graphs that are (i) non-traceable and non-hypotraceable, or (ii) in which some vertex is the centre of only spiders with more than three legs. An affirmative answer to (ii) implies an affirmative answer to (i). While non-traceable, non-hypotraceable arachnoid graphs were described in [G. Wiener, J. Graph Theory 84, No. 4, 443–459 (2017; Zbl 1359.05072)], (ii) remained open. In this paper we give an affirmative answer to this question and discuss spanning spiders whose legs must have some minimum length.

MSC:

05C45 Eulerian and Hamiltonian graphs
05C05 Trees
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