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A well-known conjecture of Babai states that if $G$ is a finite simple groupand $X$ is a generating set of $G$, then the diameter of the Cayley graph$Cay(G,X)$ is bounded above by $(\log |G|)^c$ for some absolute constant $c$.The goal of this paper is to prove such a bound for the diameter of $Cay(G,X)$whenever $G=SL(n,p)$ and $X$ is a generating set of $G$ which contains atransvection. A natural analogue of this result is also proved for $G=SL(n,K)$,where $K$ can be any field.

Diameter of Cayley graphs of $SL(n,p)$ with generating sets containing a transvection