Matheson–Tarjan domination conjecture for plane triangulations

Let $G$ be a plane triangulation of sufficiently large order $n$. A dominating set is a subset $X\subseteq V(G)$ such that $V(G)=\bigcup_{x\in X}N_G[x]$, and $\gamma(G)$ denotes the minimum cardinality of a dominating set.

Matheson–Tarjan conjecture.

$$\gamma(G)\le\frac{n}{4}.$$

Matheson and Tarjan proved the general upper bound $\gamma(G)\le n/3$ and gave examples showing that the conjectured constant would be tight; the conjecture remains open.