Recognizing Knödel graphs

Knödel graphs form a class of bipartite incident-graph of circulant digraphs. This class has been extensively studied for the purpose of fast communications in networks, and it has deserved a lot of attention in this context. In this paper, we show that there exists an O(n log⁵ n)-time algorithm to recognize Knödel graphs of order 2n. The algorithm is based on a characterization of the cycles of length six in these graphs (bipartite incident-graphs of circulant digraphs always have cycles of length six). A consequence of our result is that the circulant digraphs whose chords are the power of two minus one can be recognized in O(n log⁵ n) time.