Graphs with few hamiltonian cycles

Abstract

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