On a conjecture concerning dyadic oriented matroids.
A digraph D = (V,A) is arc-traceable if for each arc xy in A, xy lies on a directed path containing all the vertices of V, i.e., hamiltonian path. We prove a conjecture of Quintas [7]: if D is arc-traceable, then the condensation of D is a directed path. We show that the converse of this conjecture is false by providing an example of an upset tournament which is not arc-traceable. We then give a characterization for upset tournaments in terms of their score sequences, characterize which arcs of...
Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. In 1994, Sárközy and Szemerédi conjectured that there exist infinite additive complements A and B with lim sup A(x)B(x)/x ≤ 1 and A(x)B(x)-x = O(minA(x),B(x)), where A(x) and B(x) are the counting functions of A and B, respectively. We prove that, for infinite additive complements A and B, if lim sup A(x)B(x)/x ≤ 1, then, for any given M > 1,...
We assign to each positive integer a digraph whose set of vertices is and for which there is a directed edge from to if . We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.
We give a construction for regular Hadamard matrices of order where is the order of a Hadamard matrix and is the order of a regular Hadamard matrix. The construction can be used to construct regular Hadamard matrices with special properties and includes several constructions which have been given previously. In the final section we consider the case in more detail.
The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping there exists a point such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of which were previously known (as far as the author knows) only for f linear (cf. [1]).
We consider cubic graphs formed with k ≥ 2 disjoint claws (0 ≤ i ≤ k-1) such that for every integer i modulo k the three vertices of degree 1 of are joined to the three vertices of degree 1 of and joined to the three vertices of degree 1 of . Denote by the vertex of degree 3 of and by T the set . In such a way we construct three distinct graphs, namely FS(1,k), FS(2,k) and FS(3,k). The graph FS(j,k) (j ∈ 1,2,3) is the graph where the set of vertices induce j cycles (note that the graphs...