On almost hypohamiltonian graphs

A graph $G$ is almost hypohamiltonian (a.h.) if $G$ is non-hamiltonian, there exists a vertex $w$ in $G$ such that $G - w$ is non-hamiltonian, and $G - v$ is hamiltonian for every vertex $v \ne w$ in $G$. The second author asked in [J. Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here we solve this problem. To this end, we present a specialised algorithm which generates complete sets of a.h. graphs for various orders. Furthermore, we show that the smallest cubic a.h. graphs have order 26. We provide a lower bound for the order of the smallest planar a.h. graph and improve the upper bound for the order of the smallest planar a.h. graph containing a cubic vertex. We also determine the smallest planar a.h. graphs of girth 5, both in the general and cubic case. Finally, we extend a result of Steffen on snarks and improve two bounds on longest paths and longest cycles in polyhedral graphs due to Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1602.0717

Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths

In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V_1,...,V_t such that for all i the subtournament T[V_i] induced on T by V_i is strongly k-connected. Our main result implies an affirmative answer to this question. In particular we show that f(k,t) = O(k^7 t^4) suffices. As another application of our main result we give an affirmative answer to a question of Song as to whether, for any integer t, there exists an integer h(t) such that every strongly h(t)-connected tournament has a 1-factor consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t) = O(t^5) suffices.Comment: final version, to appear in Combinatoric