The Prism of the Acyclic Orientation Graph is Hamiltonian

Pruesse, Gara

doi:10.37236/1199

Cited by 8 publications

(3 citation statements)

References 2 publications

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“…This question was reiterated by Savage and Zhang [SZ98] and by Edelman (see [Sav97]). Towards this question, Squire [Squ94] showed that the square G 2 (H) has a Hamilton cycle for any graph H. Pruesse and Ruskey [PR95] strengthened this result and proved that the prism G(H) × P 2 , i.e., the Cartesian product of G(H) with a single edge, admits a Hamilton cycle. As G(H) is bipartite, this easily implies that G 2 (H) has a Hamilton cycle.…”

Section: Acyclic Orientationsmentioning

confidence: 99%

Combinatorial Gray Codes — an Updated Survey

Mütze

2023

Electron. J. Combin.

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A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605--629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

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Section: Acyclic Orientationsmentioning

confidence: 99%

Combinatorial Gray Codes — an Updated Survey

Mütze

2023

Electron. J. Combin.

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“…These acyclic reorientations posets and the underlying acyclic orientation flip graphs have been extensively studied, in particular for counting [Sta73,Las01], traversing [SSW93,PR95], and generating [Squ98,BS99] all acyclic orientations of a graph. This paper considers these acyclic reorientation posets from a lattice theoretic perspective: after characterizing the directed acyclic graphs D for which AR D is a lattice, we explore lattice properties of AR D , in particular the combinatorics and geometry of the lattice quotients of AR D when it turns out to be semidistributive.…”

Section: Introduction and Overviewmentioning

confidence: 99%

Acyclic reorientation lattices and their lattice quotients

Pilaud¹

2021

Preprint

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We prove that the acyclic reorientation poset of a directed acyclic graph D is a lattice if and only if the transitive reduction of any induced subgraph of D is a forest. We then show that the acyclic reorientation lattice is always congruence normal, semidistributive (thus congruence uniform) if and only if D is filled, and distributive if and only if D is a forest. When the acyclic reorientation lattice is semidistributive, we introduce the ropes of D that encode the join irreducibles acyclic reorientations and exploit this combinatorial model in three directions. First, we describe the canonical join and meet representations of acyclic reorientations in terms of non-crossing rope diagrams. Second, we describe the congruences of the acyclic reorientation lattice in terms of lower ideals of a natural subrope order. Third, we use Minkowski sums of shard polytopes of ropes to construct a quotientope for any congruence of the acyclic reorientation lattice.

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“…When R = ∅, AO R (G) is the acyclic orientations graph of G. AO R (G) becomes the linear extensions adjacency graph of an n element poset P when when G = K n and R and σ R are defined by the covering relations in P . In contrast to the situation for linear extensions and acyclic orientations, the square of AO R (G) is not necessarily hamiltonian.Counterexamples appear in[Squ94c] and[PR95].…”

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confidence: 99%

A Survey of Combinatorial Gray Codes

Savage¹

1997

SIAM Rev.

392

254

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The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some pre-specified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960's and 70's on minimal change listings for other combinatorial families, including permutations and combinations.The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Discrete Mathematics Conference in 1988 and his subsequent SIAM monograph in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area and most of the problems posed by Wilf are now solved.In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

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The Prism of the Acyclic Orientation Graph is Hamiltonian

Cited by 8 publications

References 2 publications

Combinatorial Gray Codes — an Updated Survey

Combinatorial Gray Codes — an Updated Survey

Acyclic reorientation lattices and their lattice quotients

A Survey of Combinatorial Gray Codes

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