We present the latest developments on the number of (≤ k)-sets and halving lines for (generalized) configurations of points; as well as the rectilinear and pseudolinear crossing numbers of Kn. In particular, we define perfect generalized configurations on n points as those whose number of (≤ k)-sets is exactly 3¡k+1 2 ¢ for all k ≤ n/3. We conjecture that for each n there is a perfect configura...

A sequence a = (a0, a1, a2, · · · , an) is said to be an almost p-ary sequence of period n + 1 if a0 = 0 and ai = ζ bi p for 1 ≤ i ≤ n, where ζp is a primitive p-th root of unity and bi ∈ {0, 1, · · · , p − 1}. Such a sequence a is called perfect if all its out-of-phase autocorrelation coefficients are zero; and is called nearly perfect if its out-of-phase autocorrelation coefficients are all 1...

For a certain graph obtained by adding extra vertices and edges to the triangular lattice graph, Propp conjectured that the number of perfect matchings of such a graph is always divisible by 3. In this note we prove this conjecture. In a graph G, a matching is a set of edges such that no two edges are incident to each other. A matching in a graph is called perfect if every vertex is incident wi...

We determine the number of perfect matchings and forests in a family Tr,3 of triangulated prism graphs. These results show that the extremal number of perfect matchings in a planar graph with n vertices is bounded below by Ω ( 6 √ 7 + √ 37 n) = Ω (1.535) and the extremal number of forests contained in a planar graph with n vertices is bounded below by

The purpose of this paper is to introduce a new approach to the problem of computing perfect matchings in fast deterministic parallel time. In particular, this approach yields a new algorithm which finds a perfect matching in bipartite cubic graphs in time O log n and O n n logn processors in the arbitrary CRCW PRAM model.

We show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair M ⊂ N of models of set theory implying that every perfect set in N has an element in N which is not in M .

Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover Q and a stable set cover S s.t. every clique in Q intersects every stable set in S. Normal graphs can be considered as closure of perfect graphs by means of co-normal products (K ̈orner [6]) and graph entropy (Czisz ́ar et al. [5]). Perfect graphs have been recently characterized a...

A near perfect matching is a matching covering all but one vertex in a graph. Let G be a connected graph and n ≤ (|V (G)|−2)/2 be a positive integer. If any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. This paper presents two construction characterizations of defect n-extendable bipartite graphs and a necessary condition for minima...

For every n = 2 k ≥ 16 there exist exactly ⌊(k + 1)/2⌋ mutually nonequiv-alent Z 4-linear extended perfect codes with distance 4. All these codes have different ranks. Codes represented in such a manner are called Z 4-linear. In [5] it is shown that the extended Golay code and the extended Hamming (n, 2 n−log 2 n−1 , 4)-codes (of length n and cardinality 2 n−log 2 n−1 , with distance 4) for eve...