Uniquely and 2-Uniquely Hamiltonian Graphs

In graph theory a Hamilton cycle is a walk around the vertices of a graph in which each vertex is visited exactly once, and then it returns to the starting vertex. The problem of determining whether a graph contains a Hamilton cycle has been studied extensively and is determined to belong to the so-called NP-complete family of problems for arbitrary graphs. Due to the difficulty in solving such a problem for an arbitrary graph, we set our sights on a family of graphs described by graph theorist John Sheehan. A maximum uniquely Hamiltonian graph contains the greatest number of edges possible while maintaining a single Hamilton cycle. Sheehan shows that for a graph with n nodes (for n \u3e= 4), the maximum number of edges it can contain is equal to (n^2/4) + 1. We will describe an algorithm that finds the Hamilton cycle for any such graph or any of its subgraphs in polynomial time. This algorithm shows that these graphs do not suffer the same complexity issues as do arbitrary graphs for a Hamilton cycle problem. For any graph containing a single Hamilton cycle, that cycle can be revealed in polynomial time

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