Hamilton-connected Mycielski \(\mathrm{graphs}^\ast\) (Q2236221)

Summary: \textit{W. Jarnicki} et al. [Ars Math. Contemp. 13, No. 2, 427--460 (2017; Zbl 1380.05066)] conjectured that if \(G\) is Hamilton-connected and not \(K_2\), then its Mycielski graph \(\mu(G)\) is Hamilton-connected. In this paper, we confirm that the conjecture is true for three families of graphs: the graphs \(G\) with \(\delta(G)>| V (G)|/2\), generalized Petersen graphs \(GP(n,2)\) and \(GP(n,3)\), and the cubes \(G^3\). In addition, if \(G\) is pancyclic, then \(\mu(G)\) is pancyclic.