On elementary lower bounds for the partition function.
On elementary moves that generate all spherical latin trades
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 entropies of occupancy distributions
On equality of edge-connectivity and minimum degree of a graph
On eulerian irregularity in graphs
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 eulerian subgraphs of complementary graphs
On Even Regular Graphs of the Third Degree
On existence of some hyplanes of order (n,m)
On expansions and extensions of powerful digraphs.
On external-memory planar depth first search.
On extremal graphs with no long paths.
On extremal sizes of locally -tree graphs
A graph is a locally -tree graph if for any vertex the subgraph induced by the neighbours of is a -tree, , where -tree is an edgeless graph, -tree is a tree. We characterize the minimum-size locally -trees with vertices. The minimum-size connected locally -trees are simply -trees. For , we construct locally -trees which are maximal with respect to the spanning subgraph relation. Consequently, the number of edges in an -vertex locally -tree graph is between and , where both...
On -domination number of a graph
On -edge cover coloring of nearly bipartite graphs.
On face-vectors and vertex-vectors of polyhedral maps on orientable -manifolds
On factorization of probability distributions over directed graphs
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
Let be a family of random independent k-element subsets of [n] = 1,2,...,n and let denote a family of ℓ-element subsets of [n] such that the event that S belongs to depends only on the edges of contained in S. Then, the edges of are ’weakly dependent’, say, the events that two given subsets S and T are in 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...
On feasible sets of mixed hypergraphs.
On Fibonacci-like sequences.
On finite and countable rigid graphs and tournaments