On The Number of Edge-Disjoint Almost Perfect Matchings in Regular Odd Order Graphs

Let \( G \) be a \( k \)-regular graph of odd order \( n \geq 3 \) with \( k \geq \frac{n + 1}{2} \). This implies that \( k \) is even. Furthermore, let
\[
p = \min\left\{\frac{k}{2}, \left\lceil k-\frac{n}{3}\right\rceil\right\}.
\]
If \( x_1, x_2, \ldots, x_p \) are arbitrary given, pairwise different, vertices of the graph \( G \), then we show in this paper that there exist \( p \) pairwise edge-disjoint almost perfect matchings \( M_1, M_2, \ldots, M_p \) in \( G \) with the property that no edge of \( M_i \) is incident with \( x_i \) for \( i = 1, 2, \ldots, p \).