Graphs having circuits with at least two chords

All 2-connected non-bipartite graphs are characterized which have a minimal valency ≥3 and which have no odd circuit with two or more chords. From this it is derived that in each graph with chromatic number ≥4 containing no complete 4-graph there is an odd circuit with two or more chords. This result was conjectured by P. Erdös. Corresponding results for even circuits and for circuits with a fixed edge are obtained. Some related problems of the Turán type are solved.