On some Aspects of my Work with Gabriel Dirac

I must have known Dirac when he was a child in the 1930's, but I really became aware of his existence when I visited England for the first time after the war in February and March 1949 . We met in London and he told me of his work on chromatic graphs . Dirac defined a k-chromatic graph to be vertex critical if the omission of any vertex decreases the chromatic number and edge critical if the removal of any edge decreases the chromatic number . I immediately liked these concepts very much and in fact felt somewhat foolish that I did not think of these natural and obviously fruitful concepts before . At that time I was already very interested in extremal problems and asked Gabriel to prove that for every k an edge critical k-chromatic graph must have o(n') edges . More precisely define fk e) (n) to be the largest integer for which there is a G(n ; fk e) (n)) (i.e . a graph on n vertices and fk e) (n) edges) which is k-chromatic and edge critical. Estimate or determine fk e) (n) as accurately as possible . Trivially f3 e) (2n + 1) = f3 e) (2n + 2) = 2n + 1 and I expected that for k > 3, fk e~(n) = o(n 2 ) . To my great surprise very soon Dirac showed [3] :