scholarly journals $k$-Cycle Free One-Factorizations of Complete Graphs

10.37236/92 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Mariusz Meszka
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Complete Graphs

It is proved that for every $n\geq 3$ and every even $k\geq 4$, where $k\neq 2n$, there exists one-factorization of the complete graph $K_{2n}$ such that any two one-factors do not induce a graph with a cycle of length $k$ as a component. Moreover, some infinite classes of one-factorizations, in which lengths of cycles induced by any two one-factors satisfy a given lower bound, are constructed.

IMPROVED LOWER BOUND FOR THE NUMBER OF KNOTTED HAMILTONIAN CYCLES IN SPATIAL EMBEDDINGS OF COMPLETE GRAPHS

2010 ◽  
Vol 19 (05) ◽  
pp. 705-708 ◽  
Author(s):  
YOSHIYASU HIRANO
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Hamiltonian Cycles ◽  
Complete Graphs

We prove that every spatial embedding of the complete graph K8 contains at least 3 knotted Hamiltonian cycles, and that every spatial embedding of Kn contains at least 3(n - 1)(n - 2) ⋯ 8 knotted Hamiltonian cycles, for n > 8.

scholarly journals Biembeddings of Metacyclic Groups and Triangulations of Orientable Surfaces by Complete Graphs

10.37236/2169 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michael John Grannell ◽  
Martin Knor
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Complete Graph ◽  
Latin Squares ◽  
Orientable Surface ◽  
Complete Graphs ◽  
Metacyclic Group ◽  
Metacyclic Groups ◽  
Multiplicative Order ◽  
Orientable Surfaces

For each integer $n\ge 3$, $n\ne 4$, for each odd integer $m\ge 3$, and for any $\lambda\in\Bbb Z_n$ of (multiplicative) order $m'$ where $m'\mid m$, we construct a biembedding of Latin squares in which one of the squares is the Cayley table of the metacyclic group $\mathbb{Z}_m\ltimes_{\lambda}\mathbb{Z}_n$. This extends the spectrum of Latin squares known to be biembeddable.The best existing lower bounds for the number of triangular embeddings of a complete graph $K_z$ in an orientable surface are of the form $z^{z^2(a-o(1))}$ for suitable positive constants $a$ and for restricted infinite classes of $z$. Using embeddings of $\mathbb{Z}_3\ltimes_{\lambda}\mathbb{Z}_n$, we extend this lower bound to a substantially larger class of values of $z$.

scholarly journals An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 525
Author(s):  
Javier Rodrigo ◽  
Susana Merchán ◽  
Danilo Magistrali ◽  
Mariló López
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
General Position ◽  
Crossing Number ◽  
Minimum Number

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

scholarly journals Minimum Number of Monotone Subsequences of Length 4 in Permutations

2014 ◽  
Vol 24 (4) ◽  
pp. 658-679 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
PING HU ◽  
BERNARD LIDICKÝ ◽  
OLEG PIKHURKO ◽  
BALÁZS UDVARI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Complete Graphs ◽  
Formula Group ◽  
Minimum Number ◽  
Additional Stability ◽  
Edge Colourings ◽  
Image Position

We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromaticK4is minimized. We show that all the extremal colourings must contain monochromaticK4only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

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.

scholarly journals On the Ramsey Number $R(4,6)$

10.37236/2102 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Fixed Point ◽  
Automorphism Group ◽  
Lower Bound ◽  
Heuristic Search ◽  
Ramsey Number ◽  
Complete Graphs ◽  
Search Procedure ◽  
Edge Colorings

The lower bound for the classical Ramsey number $R(4,6)$ is improved from 35 to 36. The author has found 37 new edge colorings of $K_{35}$ that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found using a heuristic search procedure.

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.

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.

Two-tree graphs with minimum sum-connectivity index

2020 ◽  
pp. 2150053
Author(s):  
A. Jahanbani
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Connectivity Index ◽  
Tree Graphs ◽  
Sharp Lower Bound

The sum-connectivity index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. The graphs called two-trees are defined by recursion. The smallest two-tree is the complete graph on two vertices. A two-tree on [Formula: see text] vertices (where [Formula: see text]) is obtained by adding a new vertex adjacent to the two end vertices of one edge in a two-tree on [Formula: see text] vertices. In this paper, the sharp lower bound on the sum-connectivity index of two-trees is presented, and the two-trees with the minimum and the second minimum sum-connectivity, respectively, are determined.

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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