On the existence of even and k-divisible-matchings

Abstract

The concept of an an even matching was first introduced by Billington and Hoffman. They were used to find gregarious 4-cycle decompositions of $K_{8t(a),b}$ with a and b odd. Their paper contains even matchings of type $(\alpha^8,\beta)$ for $\alpha$, $\beta$ even and $0\leq \beta\leq 4\alpha$. This paper considers the necessary and sufficient conditions for the existence of even matchings as well as k-divisible matchings. We present a construction of even matchings and 3-divisible matchings of type $(a_1,a_2,\ldots,a_p)$ provided the necessary conditions are satisfied.