an:06872397
Zbl 1387.05072
Wiener, G??bor; Zamfirescu, Carol T.
Gallai's question and constructions of almost hypotraceable graphs
EN
Discrete Appl. Math. 243, 270-278 (2018).
00401083
2018
j
05C12 05C38
Gallai's problem; traceable; hypotraceable; almost hypotraceable
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. \textit{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 \textit{C. Thomassen} [Discrete Math. 9, 91--96 (1974; Zbl 0278.05110); ibid. 14, 377--389 (1976; Zbl 0322.05130)] and \textit{L. Gargano} et al. [ibid. 285, No. 1--3, 83--95 (2004; Zbl 1044.05048)].
Zbl 0184.27504; Zbl 0278.05110; Zbl 0322.05130; Zbl 1044.05048