On recognition of alternating groups by prime graph

The prime graph $GK(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order $rs$. Let $Alt_n$ denote the alternating group of degree $n$. Assume that $p\geq13$ is a prime and $n$ is an integer such that $p\leq n\leq p+3$. We prove that if $G$ is a finite group such that $GK(G)=GK(Alt_n)$, then $G$ has a unique nonabelian composition factor, and this factor is isomorphic to $Alt_t$, where $p\leq t\leq p+3$.