Hendry, G.R.T., An Ore-type sufficient condition for a bipancyclic ordering, Discrete Mathematics 102 (1992) 47-49. It is shown that if G(X, Y, E) is a bipartite graph with 1X1= IYI = n Z= 2 in which d(x) + d(y) 2 n + 1 whenever x E X, y E Y, and xy $ E then, unless n is odd and G is one exceptional graph, G has a bipancyclic ordering, i.e. the vertices of X and Y can be labelled x,, . , x, and...

n=1 An = ∅, then lim n→∞ μ(An) = 0. (1.4) Given the ring A of subsets of Ω, a function f : Ω→ R is called A-elementary, if f(Ω) is a finite subset of R and f−1({c} ∈ A for every c ∈ R such that c 6= 0. let E = EA denote the set of all A-elementary functions on E. Provided with the pointwise addition and scalar multiplication of functions, the space F of all functions f : Ω→ R is a vector space ...

Let (X, Z) be a bivariate random vector. A predictor of X based on Z is just Borel function g(Z). The problem "least squares prediction" given the observation to find global minimum point functional E[(X − g(Z))2] with respect all variables g(Z), where g function. It well known that solution this conditional expectation E(X|Z). We also know that, if for nonnegative smooth F: R×R → R, arg ming(Z...

Best approximation results provide an approximate solution to the fixed point equation $Tx=x$, when the non-self mapping $T$ has no fixed point. In particular, a well-known best approximation theorem, due to Fan cite{5}, asserts that if $K$ is a nonempty compact convex subset of a Hausdorff locally convex topological vector space $E$ and $T:Krightarrow E$ is a continuous mapping, then there exi...

If P is a hereditary property then we show that, for the existence of a perfect f -factor, P is a sufficient condition for countable graphs and yields a sufficient condition for graphs of size א1. Further we give two examples of a hereditary property which is even necessary for the existence of a perfect f -factor. We also discuss the א2-case. We consider graphs G = (V,E), where V = V (G) is a ...

A group is a mathematical structure 〈G; ∗〉 consisting of a non-empty set G and a binary operation ∗ : G×G→ G and satisfying the following axioms: (A1) The operation ∗ is associative, i.e., for any x, y, z ∈ G, x ∗ (y ∗ z) = (x ∗ y) ∗ z. (A2) There exists a neutral element e for ∗, i.e., x ∗ e = e ∗ x = x for all x ∈ G. (A3) Every element x ∈ G has an inverse x̂, i.e., x ∗ x̂ = x̂ ∗ x = e. If ∗ is ...

Let X be a dcpo and let L be a complete lattice. The family σL(X) of all Scott continuous mappings from X to L is a complete lattice under pointwise order, we call it the L-fuzzy Scott structure on X. Let E be a dcpo. A mapping g : σL(E) −> M is called an LM-fuzzy possibility valuation of E if it preserves arbitrary unions. Denote by πLM(E) the set of all LM-fuzzy possibility valuations of E. T...

Let G be a connected graph. The vertex-set and edge-set of G denoted by V(G) and E(G) respectively. The distance between the vertices u and v, d(u,v), in a graph is the number of edges in a shortest path connecting them. Two graph vertices are adjacent if they are joined by a graph edge. The degree of a vertex ( ) i V G ∈ is the number of vertices joining to i and denoted by ( ) v i . The ( , )...

Let G be a graph and t : V ( G ) , {0, 1}, where t(V(G)) is even. (If X~_ V(G), then t ( X ) : = E { t (x):xeX}. ) A t-join is a set F ~ E ( G ) with d~(x)=t(x) (mod 2), Vx e V(G). (dF(x) denotes the number of edges of F incident with x, where loops count twice.) t-joins contain Chinese postman tours, matchings and minimum weight paths as a special case. (el. [1, 7]). If X c V(G), let 6 ( X ) =...