domatic number

• 2022/09 Tran2022
  • page 19 : bounded twin-cover number does not imply bounded domatic number – Observation 3.4. Twin Cover Number is incomparable to Maximum Clique, Domatic Number and Distance to Disconnected.
  • page 19 : bounded domatic number does not imply bounded twin-cover number – Observation 3.4. Twin Cover Number is incomparable to Maximum Clique, Domatic Number and Distance to Disconnected.
  • page 34 : bounded c-closure does not imply bounded domatic number – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
  • page 34 : bounded domatic number does not imply bounded c-closure – Observation 5.7. $c$-Closure is incomparable to Distance to Disconnected, Domatic Number and Maximum Clique.
• unknown
  • minimum degree $k$ upper bounds domatic number by $\mathcal O(k)$ – The vertex of minimum degree needs to be dominated in each of the sets. As the sets cannot overlap there can be at most $k+1$ of them.
  • graph class complete has unbounded domatic number – Parameter is unbounded for the graph class of cliques.
• https://mathworld.wolfram.com/DomaticNumber.html
  • domatic number – The maximum number of disjoint dominating sets in a domatic partition of a graph $G$ is called its domatic number $d(G)$.
• SchroderThesis
  • page 16 : bounded distance to complete does not imply bounded domatic number – Proposition 3.7
  • page 20 : bounded distance to co-cluster does not imply bounded domatic number – Proposition 3.12
  • page 20 : bounded distance to bipartite does not imply bounded domatic number – Proposition 3.12
  • page 30 : bounded bisection bandwidth does not imply bounded domatic number – Proposition 3.28
  • page 31 : bounded domatic number does not imply bounded vertex connectivity – Proposition 3.30