The number of disjoint perfect matchings in semi-regular graphs

We obtain a sharp result that for any even n ? 34, every {Dn,Dn+1}-regular graph of order n contains ?n/4? disjoint perfect matchings, where Dn = 2?n/4?-1. As a consequence, for any integer D ? Dn, every {D, D+1}- regular graph of order n contains (D-?n/4?+1) disjoint perfect matchings.

On Some Exponential Equations of S. S. Pillai

In this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, ax – by = c, where a, b and c are given nonzero integers with a, b ≥ 2. In particular, we obtain the sharp result that there are at most two solutions in positive integers x and y and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

Some Properties of Rational Functions with Prescribed Poles

Let P(z) be a polynomial of degree not exceeding n and let where |aj| > 1, j = 1, 2,…,n. If the rational function r(z) = P(z)/W(z) does not vanish in |z| < k, then for k = 1 it is known thatwhere B(Z) = W*(z)/W(z) and . In the paper we consider the case when k > 1 and obtain a sharp result. We also show thatwhere , and as a consquence of this result, we present a generalization of a theorem of O’Hara and Rodriguez for self-inversive polynomials. Finally, we establish a similar result when supremum is replaced by infimum for a rational function which has all its zeros in the unit circle.

Linear independence of translations

It was shown by Edgar and Rosenblatt that f ∈ Lp (ℝn), 1 ≤ p < 2n/ (n-1), and f ≠0, then f has linearly independent translates. Using a result of Hömander, it is shown here that the same theorem holds if p = 2n / (n−1). This gives a sharp result because for n ≥2, there exists f ∈C0 (ℝn), f ≠0, which is simultaneously in all Lp (ℝn), p > 2n/(n−1), that has a linear dependence relation among its translates. References and some discussion are included.