Finding a Longest Alternating Cycle in a 2-edge-coloured Complete Graph is in RP

1996 ◽  
Vol 5 (3) ◽  
pp. 297-306 ◽  
Author(s):  
Rachid Saad
Keyword(s):  
Complete Graph ◽  
Strong Evidence ◽  
General Setting ◽  
Maximum Length ◽  
Sufficient Condition ◽  
Complete Graphs ◽  
Random Polynomial ◽  
Exact Matching ◽  
Matching Problem ◽  
Bipartite Tournament

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

scholarly journals Popular Matchings in Complete Graphs

Algorithmica ◽  
2021 ◽  
Author(s):  
Ágnes Cseh ◽  
Telikepalli Kavitha
Keyword(s):  
Complete Graph ◽  
The Other ◽  
Complete Graphs ◽  
Matching Problem ◽  
Graph Theoretic ◽  
The Given ◽  
Popular Matching

AbstractOur input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching $$M'$$ M ′ : here each vertex casts a vote for the matching in $$\{M,M'\}$$ { M , M ′ } in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes $$\texttt {NP}$$ NP -complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and $$\texttt {NP}$$ NP -complete when n has the other parity.

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

1969 ◽  
Vol 21 ◽  
pp. 992-1000 ◽  
Author(s):  
L. W. Beineke
Keyword(s):  
Projective Plane ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Graphs ◽  
Minimum Number ◽  
Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

A study on distance in graph complement

2020 ◽  
Vol 12 (03) ◽  
pp. 2050045
Author(s):  
A. Chellaram Malaravan ◽  
A. Wilson Baskar
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The aim of this paper is to determine radius and diameter of graph complements. We provide a necessary and sufficient condition for the complement of a graph to be connected, and determine the components of graph complement. Finally, we completely characterize the class of graphs [Formula: see text] for which the subgraph induced by central (respectively peripheral) vertices of its complement in [Formula: see text] is isomorphic to a complete graph [Formula: see text], for some positive integer [Formula: see text].

scholarly journals INTRINSICALLY LINKED GRAPHS WITH KNOTTED COMPONENTS

2012 ◽  
Vol 21 (07) ◽  
pp. 1250065 ◽  
Author(s):  
THOMAS FLEMING
Keyword(s):  
Complete Graph ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Linking Number ◽  
A Minor

We construct a graph G such that any embedding of G into R3 contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a 2-component link whose linking number is a nonzero multiple of n. Finally, we show that if a graph is a Cartesian product of the form G × K2, it is intrinsically linked if and only if G contains one of K5, K3,3 or K4,2 as a minor.

Possible origins of α-damage in diamonds from kimberlite and alluvial sources

1973 ◽  
Vol 39 (303) ◽  
pp. 349-360 ◽  
Author(s):  
E. R. Vance ◽  
J. W. Harris ◽  
H. J. Milledge
Keyword(s):  
Radiation Damage ◽  
Strong Evidence ◽  
Sufficient Condition ◽  
Rock Samples ◽  
Natural Diamonds ◽  
Particle Irradiation ◽  
Submicron Scale ◽  
De Beers

SummaryHeating experiments provide strong evidence that the transparent green coats on some diamonds from each of many localities are caused by α-particle irradiation after kimberlite injection and subsequent cooling. The natural diamonds with more or less homogeneous transparent green coats, studied in this work, appear to have received doses of at least 5 × 1013−1 × 1014 α.cm−2. For Pre-Cambrian kimberlites, such doses could occur if certain regions of the diatremes contained ⩾20 ppm by weight of equivalent uranium, after kimberlite injection and solidification.Such considerations lead to the prediction that the radioelement concentrations in the Finsch kimberlite diatreme and the Bellsbank fissure kimberlite are considerably greater than those in the Premier mine, though radioelement segregation could produce the required local concentrations. Some exploratory autoradiographic measurements made on two kimberlite rock samples from Premier and De Beers Mines indicated that the radioelements were apparently distributed on a submicron scale, which would be a necessary, but not sufficient, condition for uniform α-irradiation of diamonds.Diamonds from various alluvial sources showing green and brown spots arising from much heavier and more localized radiation damage are also discussed.

scholarly journals Point Pattern Matching Algorithm for Planar Point Sets under Euclidean Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaoyun Wang ◽  
Xianquan Zhang
Keyword(s):  
Pattern Matching ◽  
Complete Graph ◽  
Point Pattern ◽  
Point Sets ◽  
Matching Problem ◽  
Matching Algorithm ◽  
Planar Point ◽  
Point Pattern Matching ◽  
Point Set ◽  
Pattern Matching Algorithm

Point pattern matching is an important topic of computer vision and pattern recognition. In this paper, we propose a point pattern matching algorithm for two planar point sets under Euclidean transform. We view a point set as a complete graph, establish the relation between the point set and the complete graph, and solve the point pattern matching problem by finding congruent complete graphs. Experiments are conducted to show the effectiveness and robustness of the proposed algorithm.

scholarly journals Perfect One-Factorization Conjecture

d'CARTESIAN ◽  
2015 ◽  
Vol 4 (1) ◽  
pp. 114
Author(s):  
Chriestie Montolalu
Keyword(s):  
Complete Graph ◽  
Complete Graphs

Perfect one-factorization of the complete graph K2n for all n greater and equal to 2 is conjectured. Nevertheless some families of complete graphs were found to have perfect one-factorization. This paper will show some of the perfect one-factorization results in some families of complete graph as well as some result in application. Keywords: complete graph, one-factorization

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