It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In this paper, we prove that digraphs of degree three and diameter k greater than or equal to 3 which miss the Moore bound by one do not exist.
Journal of Graph Theory Vol. 48, no. 2, p. 112-126