The extremal question for cycles with chords (Q2761057)

The authors prove neat extremal results concerning \(\delta =\delta (G)\geq 2\), the minimum degree of a simple finite graph \(G\) with \(n\) vertices and \(e\) edges, and chords in cycles in \(G\). (1) \(G\) has a cycle with at least \(\lceil (\delta ^2-2\delta)/2\rceil \) chords and if \(G\) has no 3-cycle and no 5-cycle then a cycle has at least \(\delta ^2-2\delta \) chords. (2) If \(n\geq 5\), \(G\) 2-connected, and \(\delta \geq 3\) then a cycle has at least 3 chords. (3) The same holds if \(\delta \geq 3\) is replaced by \(e\geq 2n-2\). (4) If \(n\geq 4\) and \(e\geq 2n-2\) then a cycle has at least two chords.