On the Minimum Number of Hamiltonian Cycles in Regular Graphs

Haythorpe, Michael

doi:10.1080/10586458.2017.1306813

Cited by 13 publications

(13 citation statements)

References 11 publications

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“…Concerning κ ≥ 4, Fleischner conjectured [13, p. 176] that every uniquely hamiltonian graph has connectivity at most 3. It seems that the first explicit construction of a 3-connected uniquely hamiltonian graph is due to Grinberg [16], and his example is of the same order (18) and only one edge larger than the smallest example there is, which was determined by Royle using a computer [26]. Aldred and Thomassen also described a 3-connected uniquely hamiltonian graph, see [19].…”

Section: Thomassen's Conjecture and A Question Of Roylementioning

confidence: 99%

Graphs with few hamiltonian cycles

2019

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We describe an algorithm for the exhaustive generation of non-isomorphic graphs with a given number k ≥ 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 ≥ 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.

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Section: Thomassen's Conjecture and A Question Of Roylementioning

confidence: 99%

Graphs with few hamiltonian cycles

2019

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“…We can also ask a similar question concerning families in which all graphs are regular. In [13] Haythorpe conjectures that for sufficiently large k, a Hamiltonian k-regular graph on n vertices contains Ω(k n ) distinct Hamiltonian cycles. It would be also be interesting to try to prove or disprove a similar lower bound for the number of distinct Hamiltonian transversals over a k-regular graph family with a Hamiltonian transversal.…”

Section: Discussionmentioning

confidence: 99%

From one to many rainbow Hamiltonian cycles

Bradshaw¹,

Halasz²,

Stacho³

2021

Preprint

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Given a graph G and a family G = {G 1 , . . . , Gn} of subgraphs of G, a transversal of G is a pair (T, φ) such that T ⊆ E(G) and φ : T → [n] is a bijection satisfying e ∈ G φ(e) for each e ∈ T . We call a transversal Hamiltonian if T corresponds to the edge set of a Hamiltonian cycle in G. We show that, under certain conditions on the maximum degree of G and the minimum degrees of the G i ∈ G, for every G which contains a Hamiltonian transversal, the number of Hamiltonian transversals contained in G is bounded below by a function of G's maximum degree. This generalizes a theorem of Thomassen stating that, for m ≥ 300, no m-regular graph is uniquely Hamiltonian. We also extend Joos and Kim's recent result that, if G = Kn and each G i ∈ G has minimum degree at least n 2 , then G has a Hamiltonian transversal: we show that, in this setting, G has exponentially many Hamiltonian transversals. Finally, we prove analogues of both of these theorems for transversals which form perfect matchings of G.

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“…Conjecture 1 (Conjecture 4.2 in [14]). For n ≥ 8, all hamiltonian 4-regular graphs of order n have at least 9 • 2 We note that in [14] neither conjecture asks for the graphs to be hamiltonian, but since various infinite families of non-hamiltonian k-regular graphs are known for every k ≥ 3among them the famous 4-regular 4-connected family described by Meredith [16]-we have added the hamiltonicity condition in the conjectures' present formulation. Conjecture 1 was recently shown not to be true by Thomassen and the author [25].…”

Section: On Two Conjectures Of Haythorpementioning

confidence: 99%

Regular graphs with few longest cycles

Zamfirescu¹

2021

Preprint

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Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant c such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly c hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput, and that it holds for 4-regular graphs of connectivity 2 with the constant 144 < c, which we believe to be minimal among all hamiltonian 4-regular graphs of sufficiently large order. We then disprove a conjecture of Haythorpe by showing that for every non-negative integer k there is a 5-regular graph on 26 + 6k vertices with 2 k+10 • 3 k+3 hamiltonian cycles. We prove that for every d ≥ 3 there is an infinite family of hamiltonian 3-connected graphs with minimum degree d, with a bounded number of hamiltonian cycles. It is shown that if a 3-regular graph G has a unique longest cycle C, at least two components of G − E(C) have an odd number of vertices on C, and that there exist 3-regular graphs with exactly two such components.

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On the Minimum Number of Hamiltonian Cycles in Regular Graphs

Cited by 13 publications

References 11 publications

Graphs with few hamiltonian cycles

Graphs with few hamiltonian cycles

From one to many rainbow Hamiltonian cycles

Regular graphs with few longest cycles

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