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On elementary moves that generate all spherical latin trades

Aleš Drápal (2009)

Commentationes Mathematicae Universitatis Carolinae

We show how to generate all spherical latin trades by elementary moves from a base set. If the base set consists only of a single trade of size four and the moves are applied only to one of the mates, then three elementary moves are needed. If the base set consists of all bicyclic trades (indecomposable latin trades with only two rows) and the moves are applied to both mates, then one move suffices. Many statements of the paper pertain to all latin trades, not only to spherical ones.

On eulerian irregularity in graphs

Eric Andrews, Chira Lumduanhom, Ping Zhang (2014)

Discussiones Mathematicae Graph Theory

A closed walk in a connected graph G that contains every edge of G exactly once is an Eulerian circuit. A graph is Eulerian if it contains an Eulerian circuit. It is well known that a connected graph G is Eulerian if and only if every vertex of G is even. An Eulerian walk in a connected graph G is a closed walk that contains every edge of G at least once, while an irregular Eulerian walk in G is an Eulerian walk that encounters no two edges of G the same number of times. The minimum length of an...

On extremal sizes of locally k -tree graphs

Mieczysław Borowiecki, Piotr Borowiecki, Elżbieta Sidorowicz, Zdzisław Skupień (2010)

Czechoslovak Mathematical Journal

A graph G is a locally k -tree graph if for any vertex v the subgraph induced by the neighbours of v is a k -tree, k ≥ 0 , where 0 -tree is an edgeless graph, 1 -tree is a tree. We characterize the minimum-size locally k -trees with n vertices. The minimum-size connected locally k -trees are simply ( k + 1 ) -trees. For k ≥ 1 , we construct locally k -trees which are maximal with respect to the spanning subgraph relation. Consequently, the number of edges in an n -vertex locally k -tree graph is between Ω ( n ) and O ( n 2 ) , where both...

On factorization of probability distributions over directed graphs

František Matúš, Bernhard Strohmeier (1998)

Kybernetika

Four notions of factorizability over arbitrary directed graphs are examined. For acyclic graphs they coincide and are identical with the usual factorization of probability distributions in Markov models. Relations between the factorizations over circuits are described in detail including nontrivial counterexamples. Restrictions on the cardinality of state spaces cause that a factorizability with respect to some special cyclic graphs implies the factorizability with respect to their, more simple,...

On families of weakly dependent random variables

Tomasz Łuczak (2011)

Banach Center Publications

Let ₙ ( k ) be a family of random independent k-element subsets of [n] = 1,2,...,n and let ( ₙ ( k ) , ℓ ) = ₙ ( k ) ( ℓ ) denote a family of ℓ-element subsets of [n] such that the event that S belongs to ₙ ( k ) ( ℓ ) depends only on the edges of ₙ ( k ) contained in S. Then, the edges of ₙ ( k ) ( ℓ ) are ’weakly dependent’, say, the events that two given subsets S and T are in ₙ ( k ) ( ℓ ) are independent for vast majority of pairs S and T. In the paper we present some results on the structure of weakly dependent families of subsets obtained in this way. We also list...

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