Factorizations Of Complete Graphs Research Articles

A Result on the Strength of Graphs by Factorizations of Complete Graphs

A Result on the Strength of Graphs by Factorizations of Complete Graphs

Generalized Hadamard Matrices and 2-Factorization of Complete Graphs

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.

Factorizations of complete graphs into cycles and 1-factors

In this paper, we consider factorizations of complete graph $K_v$ into cycles and $1$--factors. We will focus on the existence of factorizations of $K_v$ containing two nonisomorphic factors. We obtain all possible solutions for uniform factors involving $m$--cycles and $1$--factors with a few possible exceptions when $m$ is odd.

A method to construct 1‐rotational factorizations of complete graphs and solutions to the oberwolfach problem

AbstractThe concept of a 1‐rotational factorization of a complete graph under a finite group was studied in detail by Buratti and Rinaldi. They found that if admits a 1‐rotational 2‐factorization, then the involutions of are pairwise conjugate. We extend their result by showing that if a finite group admits a 1‐rotational ‐factorization with even and odd, then has at most conjugacy classes containing involutions. Also, we show that if has exactly conjugacy classes containing involutions, then the product of a central involution with an involution in one conjugacy class yields an involution in a different conjugacy class. We then demonstrate a method of constructing a 1‐rotational ‐factorization under given a 1‐rotational 2‐factorization under a finite group . This construction, given a 1‐rotational solution to the Oberwolfach problem , allows us to find a solution to when the ’s are even (), and when is an odd prime, with no restrictions on the ’s.

On factorisations of complete graphs into circulant graphs and the Oberwolfach problem

Various results on factorisations of complete graphs into circulant graphs and on 2-factorisations of these circulant graphs are proved. As a consequence, a number of new results on the Oberwolfach Problem are obtained. For example, a complete solution to the Oberwolfach Problem is given for every 2-regular graph of order 2 p where p ≡ 5 (mod 8) is prime.

Factorizations of complete graphs into brooms

Let r and n be positive integers with r < 2 n . A broom of order 2 n is the union of the path on P 2 n − r − 1 and the star K 1 , r , plus one edge joining the center of the star to an endpoint of the path. It was shown by Kubesa (2005) [10] that the broom factorizes the complete graph K 2 n for odd n and r < ⌊ n 2 ⌋ . In this note we give a complete classification of brooms that factorize K 2 n by giving a constructive proof for all r ≤ n + 1 2 (with one exceptional case) and by showing that the brooms for r > n + 1 2 do not factorize the complete graph K 2 n .

Factorizations of Complete Graphs into Trees with at most Four Non-Leave Vertices

We give a complete characterization of trees with at most four non-leave vertices, which factorize the complete graph K 2n .

Amalgamations of factorizations of complete graphs

Let t be a positive integer, and let K = ( k 1 , … , k t ) and L = ( l 1 , … , l t ) be collections of nonnegative integers. A ( t , K , L ) -factorization of a graph is a decomposition of the graph into factors F 1 , … , F t such that F i is k i -regular and l i -edge-connected. In this paper, we apply the technique of amalgamations of graphs to study ( t , K , L ) -factorizations of complete graphs. In particular, we describe precisely when it is possible to embed a factorization of K m in a ( t , K , L ) -factorization of K n .

One–factorizations of complete graphs with regular automorphism groups

Abstract The problem of the existence of one–factorizations of complete graphs which admit an automorphism group fixing some vertices and acting regularly on the others is considered.

Factorizations of complete graphs into [n, r, s, 2]-caterpillars of diameter 5 with maximum end

A tree R such that after deleting all leaves we obtain a path P is called a caterpillar. The path P is called the spine of the caterpillar R. If the spine has length 3 and R on 2n vertices contains vertices of degrees n, r, s, 2, where r, s>2, and if a vertex of degree n is an endvertex of the spine then we say that R is an [n, r, s, 2] -caterpillar with maximum end of diameter 5. We completely characterize [2k + 1, r, s, 2]-caterpillars of order 4k + 2 and diameter 5 with maximum end that factorize the complete graph K4k+2.

Atomic Latin Squares based on Cyclotomic Orthomorphisms

Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect $1$-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect $1$-factorisations of the complete graph $K_{q+1}$ for many prime powers $q$. As a result, existence of such a factorisation is shown for the first time for $q$ in $\big\{$529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 29929, 32041, 38809, 44521, 50653, 51529, 52441, 63001, 72361, 76729, 78125, 79507, 103823, 148877, 161051, 205379, 226981, 300763, 357911, 371293, 493039, 571787$\big\}$. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples.

Factorizations of complete graphs into [n, r, s, 2] -caterpillars of diameter 5 with maximum center

A tree R such that after deleting all leaves we obtain a path P is called a caterpillar. The path P is called the spine of the caterpillar R. If the spine has length 3 and R on 2n vertices contains vertices of degrees n, r, s, 2, where r, s > 2, and if a vertex of degree n is an internal vertex of the spine then we say that R is an [n, r, s, 2] -caterpillar with maximum center of diameter 5. We completely characterize [2k + 1, r, s, 2] -caterpillars of order 4k + 2 and diameter 5 with maximum center that factorize the complete graph K4k+2.

Constructing Homogeneous Factorisations of Complete Graphs and Digraphs

A homogeneous factorisation of a complete graph K n is a partition of the edge set that is invariant under a subgroup G of S n such that G is transitive on the parts of the partition and induces a vertex-transitive automorphism group on the graph corresponding to each part. A product construction is given for such factorisations.

Transversals in permutation groups and factorisations of complete graphs

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

Semirotational Tree Factorizations of Complete Graphs

We select the class of so-called semisymmetric trees and prove that every tree T from this class admits a T-factorization of a special form in the case where T is of order n = 2k ≤ 16. We formulate the conjecture that every semisymmetric tree T admits a T-factorization. We establish the existence of a T-factorization for semisymmetric trees of certain classes.

Amalgamated Factorizations of Complete Graphs

We give some sufficient conditions for an (S, U)-outline T-factorization of Kn to be an (S, U)-amalgamated T-factorization of Kn. We then apply these to give various necessary and sufficient conditions for edge coloured graphs G to have recoverable embeddings in T-factorized Kn's.

A Hill-Climbing Algorithm for the Construction of One-Factorizations and Room Squares

In this paper we describe and discuss hill-climbing algorithms for the construction of one- factorizations of complete graphs, and orthogonal one-factorizations of complete graphs (i.e., Room squares).

The divisibility theorem for isomorphic factorizations of complete graphs

The divisibility theorem for isomorphic factorizations of complete graphs

Minimizing irregularities in sports schedules using graph theory

Abstract Some combinatorial problems occurring in scheduling the games of a sports league are presented; solutions are obtained by constructing oriented factorisations of complete graphs. One considers schedules with a minimum number of breaks in the sequences of home-games and away-games and schedules with minimum number of days with breaks. Some open problems are also mentioned.