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Graphs with few Hamiltonian cycles. (English) Zbl 1429.05114
Summary: 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 [see [W. T. Tutte, J. Lond. Math. Soc. 21, 98–101 (1946; Zbl 0061.41306)]) 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 R. C. Entringer and H. Swart [J. Comb. Theory, Ser. B 29, 303–309 (1980; Zbl 0387.05017)], and sheds light on a question of H. Fleischner [J. Graph Theory 75, No. 2, 167–177 (2014; Zbl 1280.05074)] originally settled by B. Seamone [Discuss. Math., Graph Theory 35, No. 2, 207–214 (2015; Zbl 1311.05104)]. We prove equivalent formulations of the conjecture of J. A. Bondy and B. Jackson [J. Comb. Theory, Ser. B 74, No. 2, 265–275 (1998; Zbl 1026.05071)] 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 J. Sheehan [J. Graph Theory 1, 37–43 (1977; Zbl 0359.05026)] that there is no 4-regular 1H graph. Extending work of A. J. Schwenk [J. Comb. Theory, Ser. B 47, No. 1, 53–59 (1989; Zbl 0626.05038)], 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 eneralized Petersen graph G. L. Chia and C. Thomassen [Ars Comb. 104, 307–320 (2012; Zbl 1274.05254)]. Finally, complementing work of J. Sheehan [loc. cit.] 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.
##### MSC:
 05C45 Eulerian and Hamiltonian graphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05C85 Graph algorithms (graph-theoretic aspects) 05C38 Paths and cycles
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