Jerzy Wojciechowski

Sums of Connectivity Functions on Rn

A function f : R n → R is a connectivity function if the graph of its restriction f|C to any connected C ⊂ R n is connected in R n x R. The main goal of this paper is to prove that every function f : R n → R is a sum of n + 1 connectivity functions (Corollary 2.2). We will also show that if n > 1, then every function g: R n → R which is a sum of n connectivity functions is continuous on some perfect set (see Theorem 2.5) which implies that the number n + 1 in our theorem is best possible (Corollary 2.6). To prove the above results, we establish and then apply the following theorems which are of interest on their own. For every dense G δ -subset G of R n there are homeomorphisms h 1 ...,h n of R n such that R n =G∪h 1 (G)∪...∪h n (G) (Proposition 2.4). For every n >1 and any connectivity function f : R n → R, if x ∈ R n and e >0 then there exists an open set U⊂R n such that x ∈ U⊂B n (x,e), f|bd(U) is continuous, and |(x)-f(y)|

A new lower bound for Snake-in-the-Box Codes

In this paper we give a new lower bound on the length of Snake-in-the-Box Codes, i.e., induced cycles in thed-dmensional cube. The bound is a linear function of the number of vertices of the cube and improves the boundc·2d/d, wherec is a constant, proved by Danzer and Klee.

A determinant property of Catalan numbers

Abstract Catalan numbers arise in a family of persymmetric arrays with determinant 1. The demonstration involves a counting result for disjoint path systems in acyclic directed graphs.

The basis number of the powers of the complete graph

Abstract A basis of the cycle space C(G) of a graph G is h -fold if each edge of G occurs in at most h cycles of the basis. The basis number b ( G ) of G is the least integer h such that C(G) has an h -fold basis. MacLane (1937) showed that a graph G is planar if and only if b ( G )⩽2. Schmeichel (1981) proved that b ( K n )⩽3, and Banks and Schmeichel (1982) proved that b ( K 2 d )⩽4 where K 2 d is the d -dimensional hypercube. We show that b ( K n d )⩽9 for any n and d , where K n d is the cartesian d th power of the complete graph K n .

Fractional latin squares, simplex algebras, and generalized quotients

Abstract Recently the authors defined the concept of a weighted quasigroup, and showed that each weighted quasigroup is the amalgamation of a quasigroup. Similar results were obtained for symmetric and other types of quasigroups. Here we first introduce the closely related concept of a fractional latin square and show that every fractional latin square is the fractional amalgamation of a latin square; we also show that every symmetric fractional latin square is the fractional amalgamation of a symmetric latin square; and we obtain some further similar results about some other types of fractional latin squares. We then introduce the concept of a simplex algebra, and we show that our results about weighted quasigroups and fractional latin squares can be given a very natural geometric formulation in terms of projections of simplex algebras. We also show how our results can be viewed as statements concerning a generalization of the standard quotient construction defined on an algebraic system.

Extending connectivity functions on Rn

Abstract A function f : R n → R is a connectivity function if for every connected subset C of R n the graph of the restriction f↾C is a connected subset of R n+1 , and f is an extendable connectivity function if f can be extended to a connectivity function g : R n+1 → R with R n embedded into R n+1 as R n ×{0} . There exists a connectivity function f : R → R that is not extendable. We prove that for n⩾2 every connectivity function f : R n → R is extendable.

Covering the Hypercube with a Bounded Number of Disjoint Snakes

We present a construction of an induced cycle in then-dimensional hypercubeI[n] (n≥2), and a subgroup ℌn ofI[n] considered as the group ℤ2n, such that |ℌn|≤16 and the induced cycle uses exactly one element of every coset of ℌn. This proves that for anyn≥2 the vertices ofI[n] can be covered using at most 16 vertex-disjoint induced cycles.

Infinite Matroidal Version of Hall's Matching Theorem

Halls theorem for bipartite graphs gives a necessary and sufficient condition for the existence of a matching in a given bipartite graph. Aharoni and Ziv have generalized the notion of matchability to a pair of possibly infinite matroids on the same set and given a condition that is sufficient for the matchability of a given pair of finitary matroids, where the matroid is SCF (a sum of countably many matroids of finite rank). The condition of Aharoni and Ziv is not necessary for matchability. The paper gives a condition that is necessary for the existence of a matching for any pair of matroids (not necessarily finitary) and is sufficient for any pair of finitary matroids, where the matroid is SCF.

Topological dimension and sums of connectivity functions

A manually operable animated toy clacker having particular utility as a device for the display of advertising material, the device being made from sheet material such as relatively stiff paper or the like and having relatively large, opposed, flapable portions joined to a middle portion by symmetrical, oppositely curved score lines and having also animation parts attached to the flaps or to the middle portion so that, when the latter is squeezed endwise, it and the flaps move in different directions and at different rates and thereby cause the attached animation parts to move in a controlled manner to produce a uniquely interesting, always attractive and sometimes humorous effect.

On constructing snakes in powers of complete graphs

Abstract We prove the conjecture of Abbott and Katchalski that for every m ⩾ 2 there is a positive constant λm such that S(Kmnd) ⩾ λmnd − 1S(Kmd − 1) where S(Kmd) is the length of the longest snake (cycle without chords) in the cartesian product Kmd of d copies of the complete graph Km. As a corollary, we conclude that for any finite set P of primes there is a constant c = c(P) > 0 such that S(Knd) ⩾ cnd − 1 for any n divisible by an element of P and any d ⩾ 1.