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.

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 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 Hamiltonian Paths in Some Classes of Grid Graphs

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Fatemeh Keshavarz-Kohjerdi ◽  
Alireza Bagheri
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Paths ◽  
Grid Graphs ◽  
Hamiltonian Path Problem ◽  
Np Complete ◽  
Necessary And Sufficient ◽  
Linear Time Algorithms

The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and sufficient conditions for the existence of Hamiltonian paths inL-alphabet,C-alphabet,F-alphabet, andE-alphabet grid graphs. We also present linear-time algorithms for finding Hamiltonian paths in these graphs.

scholarly journals Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy

2021 ◽  
pp. 1-32
Author(s):  
P. Renjith ◽  
N. Sadagopan
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Optimization Problem ◽  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Np Hard ◽  
Polynomial Time Algorithms ◽  
Split Graphs ◽  
Np Complete ◽  
Cycle Path

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

scholarly journals Extending a perfect matching to a Hamiltonian cycle

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Adel Alahmadi ◽  
Robert E. L. Aldred ◽  
Ahmad Alkenani ◽  
Rola Hijazi ◽  
P. Solé ◽  
...  
Keyword(s):  
Graph Theory ◽  
Regular Graph ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Regular Graphs ◽  
Spanning Subgraph ◽  
International Audience ◽  
Matching Extendability

Graph Theory International audience Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.

scholarly journals On Hamiltonian Paths and Cycles in Sufficiently Large Distance Graphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Christian Löwenstein ◽  
Dieter Rautenbach ◽  
Roman Soták
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Large Distance ◽  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Distance Graph ◽  
Hamiltonian Paths ◽  
Distance Graphs ◽  
Vertex Set ◽  
International Audience

Graph Theory International audience For a positive integer n∈ℕ and a set D⊆ ℕ, the distance graph GnD has vertex set { 0,1,\textellipsis,n-1} and two vertices i and j of GnD are adjacent exactly if |j-i|∈D. The condition gcd(D)=1 is necessary for a distance graph GnD being connected. Let D={d1,d2}⊆ℕ be such that d1>d2 and gcd(d1,d2)=1. We prove the following results. If n is sufficiently large in terms of D, then GnD has a Hamiltonian path with endvertices 0 and n-1. If d1d2 is odd, n is even and sufficiently large in terms of D, then GnD has a Hamiltonian cycle. If d1d2 is even and n is sufficiently large in terms of D, then GnD has a Hamiltonian cycle.

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

APPROXIMATING THE NEAREST NEIGHBOR INTERCHARGE DISTANCE FOR NON-UNIFORM-DEGREE EVOLUTIONARY TREES

2001 ◽  
Vol 12 (04) ◽  
pp. 533-550 ◽  
Author(s):  
WING-KAI HON ◽  
TAK-WAH LAM
Keyword(s):  
Approximation Algorithms ◽  
Maximum Degree ◽  
Nearest Neighbor ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Approximation Ratio ◽  
Evolutionary Trees ◽  
Weighted Trees ◽  
Np Complete ◽  
Necessary And Sufficient

The nearest neighbor interchange (nni) distance is a classical metric for measuring the distance (dissimilarity) between evolutionary trees. It has been known that computing the nni distance is NP-complete. Existing approximation algorithms can attain an approximation ratio log n for unweighted trees and 4 log n for weighted trees; yet these algorithms are limited to degree-3 trees. This paper extends the study of nni distance to trees with non-uniform degrees. We formulate the necessary and sufficient conditions for nni transformation and devise more topology-sensitive approximation algorithms to handle trees with non-uniform degrees. The approximation ratios are respectively [Formula: see text] and [Formula: see text] for unweighted and weighted trees, where d ≥ 4 is the maximum degree of the input trees.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

scholarly journals On paths, trails and closed trails in edge-colored graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Laurent Gourvès ◽  
Adria Lyra ◽  
Carlos A. Martinhon ◽  
Jérôme Monnot
Keyword(s):  
Polynomial Time ◽  
Optimal Solution ◽  
Colored Graph ◽  
Edge Disjoint ◽  
Colored Graphs ◽  
International Audience ◽  
Np Complete ◽  
T Path ◽  
Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

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