On the Endomorphism Monoid of $K(n,4)$

Abstract

In this paper, we explicitly explore the endomorphism monoid of the circulant complete graph $K(n,4)$. We demonstrate that $Aut(K(n,4))\cong D_n$, the dihedral group of degree $n$. Furthermore, we show that $K(n,4)\cong D_n$ is unretractive for $n=4m,4m+2$ ($m\ge 2$), and that $End(K(n,4))=qEnd(K(n,4))$ and $sEnd(K(n,4))=Aut(K(n,4))$ when $n=4m,4m+2$ ($m\ge 2$). Additionally, we prove that $End(K(4m,4))$ is regular and $End(K(4m+2,4))$ is completely regular. We also solve some enumerative problems concerning $End(K(n,4))$ are solved.