Graphs with few Hamiltonian Cycles

We describe an algorithm for the exhaustive generation of non-isomorphic graphs with a given number $k \ge 0$ of hamiltonian cycles, which is especially efficient for small $k$. Our main findings, combining applications of this algorithm and existing algorithms with new theoretical results, revolve around graphs containing exactly one hamiltonian cycle (1H) or exactly three hamiltonian cycles (3H). Motivated by a classic result of Smith and recent work of Royle, we show that there exist nearly cubic 1H graphs of order $n$ iff $n \ge 18$ is even. This gives the strongest form of a theorem of Entringer and Swart, and sheds light on a question of Fleischner originally settled by Seamone. We prove equivalent formulations of the conjecture of Bondy and Jackson that every planar 1H graph contains two vertices of degree 2, verify it up to order 16, and show that its toric analogue does not hold. We treat Thomassen's conjecture that every hamiltonian graph of minimum degree at least $3$ contains an edge such that both its removal and its contraction yield hamiltonian graphs. We also verify up to order 21 the conjecture of Sheehan that there is no 4-regular 1H graph. Extending work of Schwenk, we describe all orders for which cubic 3H triangle-free graphs exist. We verify up to order $48$ Cantoni's conjecture that every planar cubic 3H graph contains a triangle, and show that there exist infinitely many planar cyclically 4-edge-connected cubic graphs with exactly four hamiltonian cycles, thereby answering a question of Chia and Thomassen. Finally, complementing work of Sheehan on 1H graphs of maximum size, we determine the maximum size of graphs containing exactly one hamiltonian path and give, for every order $n$, the exact number of such graphs on $n$ vertices and of maximum size.Comment: 29 pages; to appear in Mathematics of Computatio

Improved asymptotic upper bounds for the minimum number of pairwise distinct longest cycles in regular graphs

We study how few pairwise distinct longest cycles a regular graph can have under additional constraints. For each integer $r \geq 5$, we give exponential improvements for the best asymptotic upper bounds for this invariant under the additional constraint that the graphs are $r$-regular hamiltonian graphs. Earlier work showed that a conjecture by Haythorpe on a lower bound for this invariant is false because of an incorrect constant factor, whereas our results imply that the conjecture is even asymptotically incorrect. Motivated by a question of Zamfirescu and work of Chia and Thomassen, we also study this invariant for non-hamiltonian 2-connected $r$-regular graphs and show that in this case the invariant can be bounded from above by a constant for all large enough graphs, even for graphs with arbitrarily large girth.Comment: Submitted for publicatio

Finding long cycles in graphs

We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second follows a more standard Monte Carlo Markov Chain strategy. Special attention is devoted to Hamiltonian cycles of (non-regular) random graphs of minimal connectivity equal to three