  In 1999, De Simone and K\"{o}rner conjectured that every graph without
induced $C_5,C_7,\overline{C}_7$ contains a clique cover $\mathcal C$ and a
stable set cover $\mathcal I$ such that every clique in $\mathcal C$ and every
stable set in $\mathcal I$ have a vertex in common. This conjecture has roots
in information theory and became known as the Normal Graph Conjecture. Here we
prove that all graphs of bounded maximum degree and sufficiently large odd
girth (linear in the maximum degree) are normal. This implies that for every
fixed $d$, random $d$-regular graphs are a.a.s. normal.
