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Perfect hexagon triple systems

Abstract The graph consisting of the three 3-cycles ( a , b , c ), ( c , d , e ), and ( e , f , a ), where a , b , c , d , e , and f are distinct is called a hexagon triple. The 3-cycle ( a , c , e ) is called an “inside” 3-cycle; and the 3-cycles ( a , b , c ), ( c , d , e ), and ( e , f , a ) are called “outside” 3-cycles. A 3 k -fold hexagon triple system of order n is a pair ( X , C ), where C is an edge disjoint collection of hexagon triples which partitions the edge set of 3 kK n . Note that the outside 3-cycles form a 3 k -fold triple system. If the hexagon triple system has the additional property that the collection of inside 3-cycles ( a , c , e ) is a k -fold triple system it is said to be perfect . A perfect maximum packing of 3 kK n with hexagon triples is a triple ( X , C , L ), where C is a collection of edge disjoint hexagon triples and L is a collection of 3-cycles such that the insides of the hexagon triples plus the inside of the triangles in L form a maximum packing of kK n with triangles. This paper gives a complete solution (modulo two possible exceptions) of the problem of constructing perfect maximum packings of 3 kK n with hexagon triples.

Maximum uniformly resolvable decompositions of K v and K v - I into 3-stars and 3-cycles

Let K v denote the complete graph of order v and K v - I denote K v minus a 1-factor. In this article we investigate uniformly resolvable decompositions of K v and K v - I into r classes containing only copies of 3-stars and s classes containing only copies of 3-cycles. We completely determine the spectrum in the case where the number of resolution classes of 3-stars is maximum.

An analytical framework for self-organizing peer-to-peer anti-entropy algorithms

An analytical framework is developed for establishing exact performance measures for peer-to-peer (P2P) anti-entropy paradigms used in biologically inspired epidemic data dissemination. Major benefits of these paradigms are that they are fully distributed, self-organizing, utilize local data only via pair-wise interactions, and provide eventual consistency, reliability and scalability. We derive exact expressions for infection probabilities through elaborated counting techniques on a digraph. Considering the first passage times of a Markov chain based on these probabilities, we find the expected message delay experienced by each peer and its overall mean as a function of initial number of infectious peers. Further delay and overhead analysis is given through simulations and the analytical framework. The number of contacted peers at each round of the anti-entropy approach is an important parameter for both delay and overhead. These exact performance measures and theoretical results would be beneficial when utilizing the models in several P2P distributed system and network services such as replicated servers, multicast protocols, loss recovery, failure detection and group membership management.

Resolvable 3-star designs

Let K v be the complete graph of order v and F be a set of 1-factors of K v . In this article we study the existence of a resolvable decomposition of K v - F into 3-stars when F has the minimum number of 1-factors. We completely solve the case in which F has the minimum number of 1-factors, with the possible exception of v ? { 40 , 44 , 52 , 76 , 92 , 100 , 280 , 284 , 328 , 332 , 428 , 472 , 476 , 572 } .

The metamorphosis of λ-fold block designs with block size four into a maximum packing of λKn with 4-cycles

Abstract Let ( X , B ) be a λ -fold block design with block size four and define sets B ( C ) and E ( K 4 ⧹ C ) as follows: for each block b ∈ B , partition b into a 4-cycle and a pair of disjoint edges and place the 4-cycle in B ( C ) and the 2 disjoint edges in E ( K 4 ⧹ C ). If we can reassemble the edges belonging to E ( K 4 ⧹ C ) into a collection of 4-cycles E ( C ) with leave L , then ( X , B ( C )∪ E ( C ), L ) is a packing of λK n with 4-cycles and is called a metamorphosis of the λ -fold block design ( X , B ). In this paper we give a complete solution of the metamorphosis problem for λ -fold block designs into maximum packings of λK n with 4-cycles for all λ (with the possible exception of λ =1, n =37, and leave 2 disjoint triangles). That is, for each λ we determine the set of all n such that there exists a λ -fold block design of order n having a metamorphosis into a maximum packing of λK n with 4-cycles.

Orthogonal Trades and the Intersection Problem for Orthogonal Arrays

This work provides an orthogonal trade for all possible volumes

Exact probability distributions for peer-to-peer epidemic information diffusion

N \in \mathbb {Z}^+ \setminus \{1,2,3,4,5,7\}

Metamorphosis problems for graph designs.

N∈Z+\{1,2,3,4,5,7} for block size 4. All orthogonal trades of volume