page 8 : maximum degree $k$ upper bounds acyclic chromatic number by $k^{\mathcal O(1)}$ – Lemma 4.6 ([15]). The acyclic chromatic number $\chi_a$ is uppre bounded by the maximum degree $\Delta$ (for every graph with $\Delta > 4$). We have $\chi_a \le \Delta(\Delta-1)/2$.
page 28 : cutwidth $k$ upper bounds maximum degree by $\mathcal O(k)$ – Lemma 2.18. For any graph $G$ and any vertex $v \in V(G), cutw(g) \ge \lceil \frac{deg(v)}2 \rceil$.
page 30 : carving-width $k$ upper bounds maximum degree by $\mathcal O(k)$ – Lemma 2.20 Carving-width of a graph $G$ is at least $\Delta(G)$ where $\Delta(G)$ is the maximum degree of a graph $G$.
maximum degree $k$ upper bounds h-index by $\mathcal O(k)$ – As h-index seeks $k$ vertices of degree $k$ it is trivially upper bound by maximum degree.
bandwidth $k$ upper bounds maximum degree by $\mathcal O(k)$ – Each vertex has an integer $i$ and may be connected only to vertices whose difference from $i$ is at most $k$. There are at most $k$ bigger and $k$ smaller such neighbors.
