The Dihedral group $D_n$ acts on the complete graph 
$K_n$ naturally. This action of $D_n$ induces the 
action on the set of the 1-regular spanning subgraphs 
of the complete graph $K_n$ of even order $n$. In this 
paper we calculate the number of the equivalence classes 
of the 1-regular spanning subgraphs of the complete graph 
$K_n$ of even order $n$ by this action by using Burnside's 
Lemma. 
This problem was presented by Dr. Shun-ichiro Koh who 
is a physicist of Kochi University. Also we calculate 
the number of the equivalence classes of the maximal 
matchings of the complete graph $K_n$ with odd order $n$ 
by the group action of the Dihedral group $D_n$.