Recognizing Bipartite Incident-Graphs of Circulant Digraphs

Knödel graphs and Fibonacci graphs are two classes of bipartite incident-graph of circulant digraphs. Both graphs have been extensively studied for the purpose of fast communications in networks, and they have deserved a lot of attention in this context. In this paper, we show that there exists an O(n log n)-time algorithm to recognize Knödel graphs, and that the same technique applies to Fibonacci graphs. 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 none of the Knödel graphs are edge-transitive, apart those of 2 − 2 vertices. An open problem that arises in this field is to derive a polynomial-time algorithm for any infinite family of bipartite incident-graphs of circulant digraphs indexed by their number of vertices.