search on google journal : COLORING OF PRIME GRAPH P G$1($R$)$ and P G$2($R$)$ OF A

RING

by Sandeep S$.$ Joshi and Kishor F$.$ Pawar

then analyze The Prime Graph of P G$1($Zn$),$ get the theorem from where it started, write a more complete proof $($if necessary, write the definition, lemma that supports the theorem$)$