scholarly journals Extending Perfect Matchings to Gray Codes with Prescribed Ends

10.37236/6928 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Petr Gregor ◽  
Tomáš Novotný ◽  
Riste Škrekovski
Keyword(s):  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Hamiltonian Path ◽  
Gray Code ◽  
Computer Assisted ◽  
Gray Codes ◽  
Opposite Parity ◽  
Additional Edge ◽  
Forbidden Configurations ◽  
Binary Strings

A binary (cyclic) Gray code is a (cyclic) ordering of all binary strings of the same length such that any two consecutive strings differ in a single bit. This corresponds to a Hamiltonian path (cycle) in the hypercube. Fink showed that every perfect matching in the $n$-dimensional hypercube $Q_n$ can be extended to a Hamiltonian cycle, confirming a conjecture of Kreweras. In this paper, we study the "path version" of this problem. Namely, we characterize when a perfect matching in $Q_n$ extends to a Hamiltonian path between two prescribed vertices of opposite parity. Furthermore, we characterize when a perfect matching in $Q_n$ with two faulty vertices extends to a Hamiltonian cycle. In both cases we show that for all dimensions $n\ge 5$ the only forbidden configurations are so-called half-layers, which are certain natural obstacles. These results thus extend Kreweras' conjecture with an additional edge, or with two faulty vertices. The proof for the case $n=5$ is computer-assisted.

scholarly journals Gray codes avoiding matchings

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Darko Dimitrov ◽  
Tomáš Dvořák ◽  
Petr Gregor ◽  
Riste Škrekovski
Keyword(s):  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Hamiltonian Path ◽  
Gray Code ◽  
Gray Codes ◽  
Similar Characterization ◽  
International Audience ◽  
Forbidden Configurations ◽  
Np Complete ◽  
Necessary And Sufficient

Graphs and Algorithms International audience A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamilionicity of Q(n) with faulty edges, which is NP-complete in general, becomes polynomial for up to 2(n-1) edges provided they form a matching.

A Parallel Reduction of Hamiltonian Cycle to Hamiltonian Path in Tournaments

1995 ◽  
Vol 19 (3) ◽  
pp. 432-440 ◽  
Author(s):  
E. Bampis ◽  
M. Elhaddad ◽  
Y. Manoussakis ◽  
M. Santha
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Parallel Reduction

scholarly journals Permutational labelling of constant weight Gray codes

2002 ◽  
Vol 65 (3) ◽  
pp. 399-406
Author(s):  
Inessa Levi ◽  
Steve Seif
Keyword(s):  
Gray Code ◽  
Gray Codes ◽  
Constant Weight ◽  
Single Exception ◽  
Positive Integers

We prove that for positive integers n and r satisfying 1 < r < n, with the single exception of n = 4 and r = 2, there exists a constant weight Gray code of r-sets of Xn = {1, 2, …, n} that admits an orthogonal labelling by distinct partitions, with each subsequent partition obtained from the previous one by an application of a permutation of the underlying set. Specifically, an r-set A and a partition π of Xn are said to be orthogonal if every class of π meets A in exactly one element. We prove that for all n and r as stated, and taken modulo , there exists a list of the distinct r-sets of Xn with |Ai ∩ Ai+1| = r − 1 and a list of distinct partitions such that πi is orthogonal to both Ai and Ai+1, and πi+1 = πiλi for a suitable permutation λi of Xn.

Perfect 1-factorizations

2019 ◽  
Vol 69 (3) ◽  
pp. 479-496 ◽  
Author(s):  
Alexander Rosa
Keyword(s):  
Pairwise Disjoint ◽  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Cubic Graphs ◽  
Vertex Set ◽  
Early Survey

AbstractLetGbe a graph with vertex-setV=V(G) and edge-setE=E(G). A 1-factorofG(also calledperfect matching) is a factor ofGof degree 1, that is, a set of pairwise disjoint edges which partitionsV. A 1-factorizationofGis a partition of its edge-setEinto 1-factors. For a graphGto have a 1-factor, |V(G)| must be even, and for a graphGto admit a 1-factorization,Gmust be regular of degreer, 1 ≤r≤ |V| − 1.One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs.A 1-factorization ofGis said to beperfectif the union of any two of its distinct 1-factors is a Hamiltonian cycle ofG. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs.It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order

2019 ◽  
Vol 28 (1) ◽  
pp. 109-119
Author(s):  
Ahmad Sabri ◽  
Vincent Vajnovszki
Keyword(s):  
Positive Integer ◽  
Gray Code ◽  
Lexicographic Order ◽  
Gray Codes ◽  
Fixed Length ◽  
Generating Algorithms

Abstract A generalized (resp. p-ary) ballot sequence is a sequence over the set of non-negative integers (resp. integers less than p) where in any of its prefixes each positive integer i occurs at most as often as any integer less than i. We show that the Reected Gray Code order induces a cyclic 3-adjacent Gray code on both, the set of fixed length generalized ballot sequences and p-ary ballot sequences when p is even, that is, ordered list where consecutive sequences (regarding the list cyclically) differ in at most 3 adjacent positions. Non-trivial efficient generating algorithms for these ballot sequences, in lexicographic order and for the obtained Gray codes, are also presented.

scholarly journals Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

10.37236/9143 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Marién Abreu ◽  
John Baptist Gauci ◽  
Domenico Labbate ◽  
Giuseppe Mazzuoccolo ◽  
Jean Paul Zerafa
Keyword(s):  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Complete Bipartite Graph ◽  
Line Graph ◽  
Sufficient Conditions ◽  
Perfect Matchings ◽  
Hamiltonian Cycles ◽  
Hamiltonian Graph ◽  
Open Problems ◽  
Hamiltonian Property

A graph admitting a perfect matching has the Perfect–Matching–Hamiltonian property (for short the PMH–property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices, or (iv) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.

scholarly journals Binary Gray Codes with Long Bit Runs

10.37236/1720 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Luis Goddyn ◽  
Pavol Gvozdjak
Keyword(s):  
Hamming Distance ◽  
Gray Code ◽  
Gray Codes

We show that there exists an $n$-bit cyclic binary Gray code all of whose bit runs have length at least $n - 3\log_2 n$. That is, there exists a cyclic ordering of $\{0,1\}^n$ such that adjacent words differ in exactly one (coordinate) bit, and such that no bit changes its value twice in any subsequence of $n-3\log_2 n$ consecutive words. Such Gray codes are 'locally distance preserving' in that Hamming distance equals index separation for nearby words in the sequence.

scholarly journals Eigenvalues, traceability, and perfect matching of a graph

2021 ◽  
Vol 57 (3) ◽  
pp. 274-285
Author(s):  
V. I. Benediktovich
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Perfect Matching ◽  
Hamiltonian Path ◽  
Recognition Algorithm ◽  
Recognition Problem ◽  
Cyclomatic Number ◽  
Np Complete

It is well known that the recognition problem of the existence of a perfect matching in a graph, as well as the recognition problem of its Hamiltonicity and traceability, is NP-complete. Quite recently, lower bounds for the size and the spectral radius of a graph that guarantee the existence of a perfect matching in it have been obtained. We improve these bounds, firstly, by using the available bounds for the size of the graph for existence of a Hamiltonian path in it, and secondly, by finding new lower bounds for the spectral radius of the graph that are sufficient for the traceability property. Moreover, we develop the recognition algorithm of the existence of a perfect matching in a graph. This algorithm uses the concept of a (κ,τ)-regular set, which becomes polynomial in the class of graphs with a fixed cyclomatic number.

scholarly journals A Less Complex Algorithmic Procedure for Computing Gray Codes

2009 ◽  
Vol 6 (2) ◽  
pp. 12 ◽  
Author(s):  
Afaq Ahmad ◽  
Mohammed M. Bait Suwailam
Keyword(s):  
Boolean Function ◽  
Gray Code ◽  
Gray Codes ◽  
Time And Space ◽  
Design And Implementation ◽  
Algorithmic Procedure

 The purpose of this paper is to present a new and faster algorithmic procedure for generating the n bi Gray codes. Thereby, through this paper we have presented the derivation, design and implementation of a newly developed algorithm for the generation of an n-bit binary reflected Gray code sequences. The developed algorithm is stemmed from the fact of generating and properly placing the min-terms from the universal set of all the possible min-terms [m0 m1 m2 …. mN] of Boolean function of n variables, where, 0 < N <  2n-1. The resulting algorithm is in concise form and trivial to implement. Furthermore, the developed algorithm is equipped with added attributes of optimizing of time and space while executed. 

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