71 research outputs found
Shortening array codes and the perfect 1-factorization conjecture
The existence of a perfect 1-factorization of the complete graph with n nodes, namely, K_n , for arbitrary even number n, is a 40-year-old open problem in graph theory. So far, two infinite families of perfect 1-factorizations have been shown to exist, namely, the factorizations of K_(p+1) and K_2p , where p is an arbitrary prime number (p > 2) . It was shown in previous work that finding a perfect 1-factorization of K_n is related to a problem in coding, specifically, it can be reduced to constructing an MDS (Minimum Distance Separable), lowest density array code. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the K_(p+1) family of perfect 1-factorization from the K_2p family. Namely, techniques from coding theory are used to prove a new result in graph theory-that the two factorization families are related
Perfect 1-factorisations of circulants with small degree
A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant graphs that admit a perfect 1-factorisation and we solve the existence problem for a large family of 4-regular circulants. Results of computer searches for perfect 1-factorisations of 4-regular circulant graphs of orders up to 30 are provided and some problems are posed
The existence of Ck-factorizations of K2n − F
AbstractA necessary condition for the existence of a Ck-factorization of K2n − F is that k divides 2n. It is known that neither K6 − F nor K12 − F admit a C3-factorization. In this paper we show that except for these two cases, the necessary condition is also sufficient
A constructive solution to the Oberwolfach Problem with a large cycle
For every -regular graph of order , the Oberwolfach problem
asks whether there is a -factorization of ( odd) or minus a
-factor ( even) into copies of . Posed by Ringel in 1967 and
extensively studied ever since, this problem is still open. In this paper we
construct solutions to whenever contains a cycle of length greater
than an explicit lower bound. Our constructions combine the
amalgamation-detachment technique with methods aimed at building
-factorizations with an automorphism group having a nearly-regular action on
the vertex-set.Comment: 11 page
A survey on constructive methods for the Oberwolfach problem and its variants
The generalized Oberwolfach problem asks for a decomposition of a graph
into specified 2-regular spanning subgraphs , called factors.
The classic Oberwolfach problem corresponds to the case when all of the factors
are pairwise isomorphic, and is the complete graph of odd order or the
complete graph of even order with the edges of a -factor removed. When there
are two possible factor types, it is called the Hamilton-Waterloo problem.
In this paper we present a survey of constructive methods which have allowed
recent progress in this area. Specifically, we consider blow-up type
constructions, particularly as applied to the case when each factor consists of
cycles of the same length. We consider the case when the factors are all
bipartite (and hence consist of even cycles) and a method for using circulant
graphs to find solutions. We also consider constructions which yield solutions
with well-behaved automorphisms.Comment: To be published in the Fields Institute Communications book series.
23 pages, 2 figure
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