The Planar Turán Number of ${\Theta }_6$-graphs

Abstract

There are two particular ${\Theta }_6$-graphs - the 6-cycle graphs with a diagonal. We find the planar Turán number of each of them, i.e. the maximum number of edges in a planar graph $G$ of $n$ vertices not containing the given ${\Theta }_6$ as a subgraph and we find infinitely many extremal constructions showing the sharpness of these results - apart from a small additive constant error in one of the cases.