TITLE: Graph in which no cycle has two crossing chords
QUESTION [13 upvotes]: Let $G$ be a graph which does not contain a simple cycle $v_1\ldots v_k$ and two "crossing" chords $v_iv_j$ and $v_pv_q$, $i<p<j<q$. An example of such graph is a triangulation of the convex polygon. Is it true that the number of edges in $G$ does not exceed $2n-3$, where $n$ denotes the number of vertices?

REPLY [12 votes]: Thomassen and Toft [JCTB 31(2):199-224, 1981] showed that any graph with minimum degree at least 3 contains a cycle with two crossing chords from neighbouring vertices on the cycle. The $2n-3$ upper bound follows by induction on $n$, since we may delete a vertex of degree at most $2$.