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Gallai’s question and constructions of almost hypotraceable graphs. (English) Zbl 1387.05072
Summary: Consider a graph \(G\) in which the longest path has order \(| V(G) | - 1\). We denote the number of vertices \(v\) in \(G\) such that \(G - v\) is non-traceable with \(\mathfrak{t}_G\). Gallai asked in 1966 whether, in a connected graph, the intersection of all longest paths is non-empty. H. Walther [J. Comb. Theory 6, 1–6 (1969; Zbl 0184.27504)] showed that, in general, this is not true. In a graph \(G\) in which the longest path has \(| V(G) | - 1\) vertices, the answer to Gallai’s question is positive iff \(\mathfrak{t}_G \neq 0\). In this article we study almost hypotraceable graphs, which constitute the extremal case \(\mathfrak{t}_G = 1\). We give structural properties of these graphs, establish construction methods for connectivities 1 through 4, show that there exists a cubic 3-connected such graph of order 28, and draw connections to works of C. Thomassen [Discrete Math. 9, 91–96 (1974; Zbl 0278.05110); ibid. 14, 377–389 (1976; Zbl 0322.05130)] and L. Gargano et al. [ibid. 285, No. 1–3, 83–95 (2004; Zbl 1044.05048)].

MSC:
05C12 Distance in graphs
05C38 Paths and cycles
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