scholarly journals Generalized Hadamard Matrices and 2-Factorization of Complete Graphs

2020 ◽  
pp. 144-151
Author(s):  
W. V. Nishadi ◽  
A. A. I. Perera
Keyword(s):  
Cyclic Group ◽  
Complete Graph ◽  
Hadamard Matrix ◽  
Edge Coloring ◽  
Hadamard Matrices ◽  
Hamiltonian Cycles ◽  
Complete Graphs ◽  
Spanning Subgraph ◽  
Generalized Hadamard Matrix ◽  
Conjugate Transpose

Graph factorization plays a major role in graph theory and it shares common ideas in important problems such as edge coloring and Hamiltonian cycles. A factor  of a graph  is a spanning subgraph of  which is not totally disconnected. An - factor is an - regular spanning subgraph of  and  is -factorable if there are edge-disjoint -factors  such that . We shall refer as an -factorization of a graph . In this research we consider -factorization of complete graph. A graph with  vertices is called a complete graph if every pair of distinct vertices is joined by an edge and it is denoted by . We look into the possibility of factorizing  with added limitations coming in relation to the rows of generalized Hadamard matrix over a cyclic group. Over a cyclic group  of prime order , a square matrix  of order  all of whose elements are the  root of unity is called a generalized Hadamard matrix if , where  is the conjugate transpose of matrix  and  is the identity matrix of order . In the present work, generalized Hadamard matrices over a cyclic group  have been considered. We prove that the factorization is possible for  in the case of the limitation 1, namely, If an edge  belongs to the factor , then the and  entries of the corresponding generalized Hadamard matrix should be different in the   row. In Particular,  number of rows in the generalized Hadamard matrices is used to form -factorization of complete graphs. We discuss some illustrative examples that might be used for studying the factorization of complete graphs.

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 On Colorful Edge Triples in Edge-Colored Complete Graphs

2020 ◽  
Vol 36 (6) ◽  
pp. 1623-1637
Author(s):  
Gábor Simonyi
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Complete Graphs ◽  
Edge Colorings ◽  
Minimum Number ◽  
Ramsey Type

Abstract An edge-coloring of the complete graph $$K_n$$ K n we call F-caring if it leaves no F-subgraph of $$K_n$$ K n monochromatic and at the same time every subset of |V(F)| vertices contains in it at least one completely multicolored version of F. For the first two meaningful cases, when $$F=K_{1,3}$$ F = K 1 , 3 and $$F=P_4$$ F = P 4 we determine for infinitely many n the minimum number of colors needed for an F-caring edge-coloring of $$K_n$$ K n . An explicit family of $$2\lceil \log _2 n\rceil $$ 2 ⌈ log 2 n ⌉ 3-edge-colorings of $$K_n$$ K n so that every quadruple of its vertices contains a totally multicolored $$P_4$$ P 4 in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the cycle of length 5.

scholarly journals Restricted Extension of Sparse Partial Edge Colorings of Complete graphs

10.37236/9552 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Carl Johan Casselgren ◽  
Lan Anh Pham
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Positive Answer ◽  
Complete Graphs ◽  
Edge Colorings ◽  
Proper Edge Coloring

Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

scholarly journals Large Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition

10.37236/7049 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
András Gyárfás ◽  
Gábor Sárközy
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Complete Graphs ◽  
Connected Component ◽  
Degree Condition ◽  
Colored Graphs ◽  
Spanning Subgraph ◽  
Delta G

It is well-known that in every $k$-coloring of the edges of the complete graph $K_n$ there is a monochromatic connected component of order at least ${n\over k-1}$. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For $k=2$ the authors proved that $\delta(G)\ge{3n\over 4}$ ensures a monochromatic connected component with at least $\delta(G)+1$ vertices in every $2$-coloring of the edges of a graph $G$ with $n$ vertices. This result is sharp, thus for $k=2$ we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is  that for larger values of $k$ the situation is different, graphs of minimum degree $(1-\epsilon_k)n$ can replace complete graphs and still there is a monochromatic connected component of order at least ${n\over k-1}$, in fact $$\delta(G)\ge \left(1 - \frac{1}{1000(k-1)^9}\right)n$$ suffices.Our second result is an improvement of this bound for $k=3$. If the edges of $G$ with  $\delta(G)\geq {9n\over 10}$ are $3$-colored, then there is a monochromatic component of order at least ${n\over 2}$. We conjecture that this can be improved to ${7n\over 9}$ and for general $k$ we conjecture the following: if $k\geq 3$ and  $G$ is a graph of order $n$ such that $\delta(G)\geq \left( 1 - \frac{k-1}{k^2}\right)n$, then in any $k$-coloring of the edges of $G$ there is a monochromatic connected component of order at least ${n\over k-1}$.

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 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.

HADAMARD MATRICES AND 1‐FACTORIZATIONS OF COMPLETE GRAPHS

Mathematika ◽  
2019 ◽  
Vol 65 (3) ◽  
pp. 488-499
Author(s):  
Keith Ball ◽  
Oscar Ortega‐Moreno ◽  
Maria Prodromou
Keyword(s):  
Hadamard Matrices ◽  
Complete Graphs

Colorful Partitions of Cardinal Numbers

1979 ◽  
Vol 31 (3) ◽  
pp. 524-541
Author(s):  
J. Baumgartner ◽  
P. Erdös ◽  
F. Galvin ◽  
J. Larson
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Cardinal Numbers ◽  
Symmetrizable Space ◽  
Element Set

Use the two element subsets of κ, denoted by [κ]2, as the edge set for the complete graph on κ points. Write CP(κ, µ, v) if there is an edge coloring R: [κ]2 → µ with µ colors so that for every proper v element set X ⊊ κ, there is a point x ∈ κ ∼ X so that the edges between x and X receive at least the minimum of µ and v colors. Write CP⧣(K, µ, v) if the coloring is oneto- one on the edges between x and elements of X.Peter W. Harley III [5] introduced CP and proved that for κ ≧ ω, CP(κ+, κ, κ) holds to solve a topological problem, since the fact that CP(ℵ1, ℵ0, ℵ0) holds implies the existence of a symmetrizable space on ℵ1 points in which no point is a Gδ.

scholarly journals Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

2018 ◽  
Vol 6 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Takuya Ikuta ◽  
Akihiro Munemasa
Keyword(s):  
Association Scheme ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Association Schemes ◽  
Roots Of Unity ◽  
Galois Rings ◽  
Type Complex ◽  
Complex Hadamard Matrix

Abstract We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

scholarly journals Group properties of Hadamard matrices

1976 ◽  
Vol 21 (2) ◽  
pp. 247-256 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Symmetric Design ◽  
Image Position ◽  
Square Matrix

An Hadamard matrix H is a square matrix of order n all of whose entries are ± 1 such thatThere are matrices of order 1 and 2and for all other Hadamard matrices the order n is a multiple of 4, n = 4m. It is a reasonable conjecture that Hadamard matrices exist for every order which is a multiple of 4 and the lowest order in doubt is 268. With every Hadamard matrix H4m a symmetric design D exists with

Sign in / Sign up

Export Citation Format

Share Document