page 9 : boxicity $k$ upper bounds chordality by $\mathcal O(k)$ – Lemma 4.15. The boxicity $b$ upper-bounds the chordality $c$. We have $c \le b$.
page 9 : distance to interval $k$ upper bounds boxicity by $\mathcal O(k)$ – Lemma 4.16. The distance $i$ to an interval graph upper bounds the boxicity $b$. We have $b \le i+1$.