A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. An old conjecture states that for every fixed k, the maximum number of edges in a k-quasi-planar graph on n vertices is O(n). Fox and Pach showed that every kquasi-planar graph with n vertices and no pair of edges intersecting in more than O(1) points has at most n( log n log k ) k) edges. We improve this up...

It is proved that the quasi-proximity space induced by the bicompletion of a quasi-uniform T0-space X is a subspace of the quasi-proximity space induced by the Samuel bicompactification of X. The result is then used to establish that the locally finite covering quasi-uniformity defined on the category Top0 of topological T0-spaces and continuous maps is not lower K-true (in the sense of Brümmer...

The object of this paper is to study (k, μ)-paracontact metric manifolds with qusi-conformal curvature tensor. It has been shown that, h-quasi conformally semi-symmetric and φ-quasi-conformally semi-symmetric (k, μ)-paracontact metric manifold with k 6= −1 cannot be an η-Einstein manifold.

A connected graph G = (V, E) is called a quasi-tree graph if there exists a vertex u0 ∈ V (G) such that G−u0 is a tree. A connected graph G = (V, E) is called a quasi-unicyclic graph if there exists a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph. Set T (n, k) := {G : G is a n-vertex quasi-tree graph with k pendant vertices}, and T (n, d0, k) := {G : G ∈ T (n, k) and there is a vertex ...

A connected graph G = (V, E) is called a quasi-tree graph if there exists a vertex u0 ∈ V (G) such that G−u0 is a tree. A connected graph G = (V, E) is called a quasi-unicyclic graph if there exists a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph. Set T (n, k) := {G : G is a n-vertex quasi-tree graph with k pendant vertices}, and T (n, d0, k) := {G : G ∈ T (n, k) and there is a vertex ...

In part I it was shown that for each k ≥ 1 the generalized Sato–Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient Gk of its fundamental group, ‘functorially’ invariant under k-quasi-isotopy. Here we show that Cochran’s derived invariant β, provided k ≥ 3, and a series of μ̄-invariants, starting with μ̄(111112122) ...

The quasi-random theory for graphs mainly focuses on a large equivalent class of graph properties each of which can be used as a certificate for randomness. For k-graphs (i.e., k-uniform hypergraphs), an analogous quasi-random class contains various equivalent graph properties including the k-discrepancy property (bounding the number of edges in the generalized induced subgraph determined by an...

It is shown that for each k > 1, if f is a Baire one function and f is the product of k bounded Darboux (quasi–continuous) functions, then f is the product of k bounded Darboux (quasi–continuous) Baire one functions as well.