Excluding cycles with a fixed number of chords

Trotignon and Vušković completely characterized graphs that do not contain cycles with exactly one chord. In particular, they show that such a graph G has chromatic number at most max(3, ω(G)). We generalize this result to the class of graphs that do not contain cycles with exactly two chords and the class of graphs that do not contain cycles with exactly three chords. More precisely we prove that graphs with no cycle with exactly two chords have chromatic number at most 6. And a graph G with no cycle with exactly three chords have chromatic number at most max(96, ω(G) + 1).