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We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe's computational enumerative results from [Experim. Math. 27 (2018) 426-430]. Thereafter, we use the Lov\'asz Local Lemma to extend Thomassen's independent dominating set method. Regarding the limitations of this method, we answer a question of Haxell, Seamone, and Verstraete, and settle the first open case of a problem of Thomassen. Motivated by an observation of Aldred and Thomassen, we prove that for every $\kappa \in \{ 2, 3 \}$ and any positive integer $k$, there are infinitely many non-regular graphs of connectivity $\kappa$ containing exactly one hamiltonian cycle and in which every vertex has degree $3$ or $2k$.

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This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian completion problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in the context of various specific kinds of undirected graphs (e.g. trees, unicyclic graphs and series-parallel graphs). The proposed algorithm, however, concentrates on solving HCP for general undirected graphs. It can be considered to belong to the category of matheuristics, because it integrates an exact linear time solution for trees into a local search algorithm for general graphs. This integration makes use of the close relation between HCP and the minimum path partition problem, which makes the algorithm equally useful for solving the latter problem. Furthermore, a benchmark set of problem instances is constructed for demonstrating the quality of the proposed algorithm. A comparison with state-of-the-art solvers indicates that the proposed algorithm is able to achieve high-quality results.

We prove that if an $n$-vertex graph with minimum degree at least $3$ contains a Hamiltonian cycle, then it contains another cycle of length $n-o(n)$; this implies, in particular, that a well-known conjecture of Sheehan from 1975 holds asymptotically. Our methods, which combine constructive, poset-based techniques and non-constructive, parity-based arguments, may be of independent interest.

In 2012 we announced “the House of Graphs” (https://houseofgraphs.org) (Brinkmann et al. 2013), which was a new database of graphs. The House of Graphs hosts complete lists of graphs of various graph classes, but its main feature is a searchable database of so called “interesting” graphs, which includes graphs that already occurred as extremal graphs or as counterexamples to conjectures. An important aspect of this database is that it can be extended by users of the website.
Over the years, several new features and graph invariants were added to the House of Graphs and users uploaded many interesting graphs to the website. But as the development of the original House of Graphs website started in 2010, the underlying frameworks and technologies of the website became outdated. This is why we completely rebuilt the House of Graphs using modern frameworks to build a maintainable and expandable web application that is future-proof. On top of this, several new functionalities were added to improve the application and the user experience.
This article describes the changes and new features of the new House of Graphs website.

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 [European J. Combin., 97 (2021), 103395], 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 nonnegative integer k there is a 5-regular graph on 26 + 6k vertices with 2^(k+10) \cdot 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.

The construction of complete lists of regular graphs up to isomorphism is one of the oldest problems in constructive combinatorics. In this article an efficient algorithm to generate regular graphs with a given number of vertices and vertex degree is introduced. The method is based on orderly generation refined by criteria to avoid isomorphism checking and combined with a fast test for canonicity. The implementation allows computing even large classes of graphs, like construction of the 4-regular graphs on 18 vertices and, for the first time, the 5-regular graphs on 16 vertices. Also in cases with given girth, some remarkable results are obtained. For instance, the 5-regular graphs with girth 5 and minimal number of vertices were generated in less than 1 h. There exist exactly four (5, 5)-cages. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 137–146, 1999

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.

A graph construction that produces a k-regular graph on n vertices for any choice of k >= 3 and n = m(k+1) for integer m >= 2 is described. The number of Hamiltonian cycles in such graphs can be explicitly determined as a function of n and k, and empirical evidence is provided that suggests that this function gives a tight upper bound on the minimum number of Hamiltonian cycles in k-regular graphs on n vertices for k >= 5 and n >= k + 3. An additional graph construction for 4-regular graphs is described for which the number of Hamiltonian cycles is superior to the above function in the case when k = 4 and n >= 11.

This paper seeks to review some ideas and results relating to Hamiltonian graphs. We list the well known results which are to be found in most undergraduate graph theory courses and then consider some old theorems which are fundamental to planar graphs. By restricting our attention to 3-connected cubic planar graphs (a class of graphs of interest to Four Colour Theorists), we are able to report on recent results regarding the smallest nonHamiltonian graphs. We then consider regular graphs generally and what might be said about when the number of Hamiltonian cycles is greater than one. Another interesting class of graphs are the bipartite graphs. In general these are not Hamiltonian but there is a famous conjecture due to Barnette that suggests that 3-connected cubic bipartite planar graphs are Hamiltonian. In our final two sections we consider this along with another open conjecture due to Barnette.

This paper explores a dynamic programming approach to the solution of three sequencing problems: a scheduling problem involving arbitrary cost functions, the traveling-salesman problem, and an assembly line balancing problem. Each of the problems is shown to admit of numerical solution through the use of a simple recursion scheme; these recursion schemes also exhibit similarities and contrasts in the structures of the three problems. For large problems, direct solution by means of dynamic programming is not practical, but procedures are given for obtaining good approximate results by solving sequences of smaller derived problems. Experience with a computer program for the solution of traveling-salesman problems is presented.

We construct an infinite family of uniquely hamiltonian graphs of minimum degree 4, maximum degree 14, and of arbitrarily high maximum degree.

The fact that a cubic hamiltonian graph must have at least three spanning cycles suggests the question of whether every hamiltonian graph in which each point has degree at least 3 must have at least three spanning cycles. We answer this in the negative by exhibiting graphs on n=2m+1, m≥5, points in which one point has degree 4, all others have degree 3, and only two spanning cycles exist.

This chapter describes Hamiltonian cycles and uniquely edge colorable graphs. If G is uniquely edge colorable then the subgraph induced by the edges of two given colors is connected and so, is a path or a cycle. A theorem is presented that states that in any cubic graph, the number of Hamiltonian cycles containing a given edge is even. If the graph is cubic and bipartite, a theorem of Kotzig tells that the total number of Hamiltonian cycles in the graph is even too. These two theorems are in fact consequences of a more general result. The chapter also presents the sets of edge-disjoint Hamiltonian cycles in multigraphs.

We report the current state of the graph isomorphism problem from the
practical point of view. After describing the general principles of the
refinement-individualization paradigm and proving its validity, we explain how
it is implemented in several of the key programs. In particular, we bring the
description of the best known program nauty up to date and describe an
innovative approach called Traces that outperforms the competitors for many
difficult graph classes. Detailed comparisons against saucy, Bliss and conauto
are presented.

We describe a general sufficient condition for a Hamiltonian graph to contain another Hamiltonian cycle. We apply it to prove that every longest cycle in a 3-connected cubic graph has a chord. We also verify special cases of an old conjecture of Sheehan on Hamiltonian cycles in 4-regular graphs and a recent conjecture on a second Hamiltonian cycle by Triesch, Nolles, and Vygen.

Abstract A graph is uniquely hamiltonian if it contains exactly one hamiltonian cycle. In this note we prove that there are no r-regular uniquely hamiltonian graphs when r > 22. This improves upon earlier results of Thomassen. Key words: C-independent set, Lov¶asz Local Lemma, Uniquely hamiltonian

In 1975, John Sheehan conjectured that every Hamiltonian 4-regular graph has a second Hamiltonian cycle. Combined with earlier results this would imply that every Hamiltonianr-regular graph (r⩾3) has a second Hamiltonian cycle. We shall verify this forr⩾300.

A note on independent dominating sets and second hamiltonian cycles

- M Ghandehari
- H Hatami

M. Ghandehari and H. Hatami. A note on independent dominating sets and second
hamiltonian cycles. Manuscript.

Code and certificates for the paper "Few hamiltonian cycles in graphs with one or two vertex degrees

- J Goedgebeur
- J Jooken
- O.-H S Lo
- B Seamone
- C T Zamfirescu

J. Goedgebeur, J. Jooken, O.-H. S. Lo, B. Seamone, and C. T. Zamfirescu. Code and
certificates for the paper "Few hamiltonian cycles in graphs with one or two vertex
degrees" (2022). See: https://github.com/JorikJooken/hamiltonian_cycles

K 2 -Hamiltonian Graphs: II

- J Goedgebeur
- J Renders
- G Wiener
- C T Zamfirescu

J. Goedgebeur, J. Renders, G. Wiener, and C. T. Zamfirescu. K 2 -Hamiltonian
Graphs: II. Submitted for publication.

The smallest uniquely hamiltonian graph with minimum degree at least

- G Royle

G. Royle. The smallest uniquely hamiltonian graph with minimum degree
at
least
3
(2017).
https://mathoverflow.net/questions/255784/
what-is-the-smallest-uniquely-hamiltonian-graph-with-minimum-degree-at-least-3/