On the mutual embeddability of (2k, k, k − 1) and (2k − 1, k, k) quasi-residual designs

On the coexistence of conference matrices and near resolvable 2-(2k+1, k, k-1) designs

We show that a near resolvable 2-(2k + 1, k, k − 1) design exists if and only if a conference matrix of order 2k+2 does. A known result on conference matrices then allows us to conclude that a near resolvable 2-(2k + 1, k, k − 1) design with even k can only exist if 2k + 1 is the sum of two squares. In particular, neither a near resolvable 2-(21, 10, 9) design nor a near resolvable 2-(33, 16, 1...

Optimal Consecutive-k-out-of-(2k+1): G Cycle

We present a complete proof for the invariant optimal assignment for consecutive-k-outof-(2k+1):G cycle, which was proposed by Zuo and Kao in 1990 with an incomplete proof, pointed out recently by Jalali, Hawkes, Cui and Hwang.

On the Rational Recursive Sequence X_{N+1}=ƔX_{N-K}+(AX_N+BX_{N-K})⁄(CX_N-DX_{N-K})

The (2k-1)-connected multigraphs with at most k-1 disjoint cycles

In 1963, Corrádi and Hajnal proved that for all k≥1 and n≥3k, every (simple) graph G on n vertices with minimum degree δ(G)≥2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k− 1)-connected multigraphs do not contain k disjoint cycles? Recently, the authors characterized the s...

On the family of Diophantine triples { k − 1 , k + 1 , 16 k 3 − 4 k }

It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any two distinct elements of the set {k − 1, k + 1, 16k − 4k, d} increased by 1 is a perfect square, then d = 4k or d = 64k−48k+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k − 1, k + 1, c, d} are regular.