On induced subgraphs with degree parity conditions in Paley graphs and Paley tournaments

Abstract: In this paper, we investigate the number of induced subgraphs and subdigraphs of Paley graphs and Paley tournaments where the (out-)degree of each vertex has the same parity. For Paley graphs, we establish a lower bound for the number of large even induced subgraphs, particularly those containing a constant proportion of vertices. We determine the number of even-even partitions of Paley graphs, showing it is exponential if $q\equiv 1\Mod{8}$ and is trivial if $q\equiv 5\Mod{8}$, while proving the non-existence of even-even partition for Paley tournaments. Furthermore, we derive asymptotic formulas for the numbers of even induced sub(di)graphs of order $r=o(q^{1/4})$ in Paley graphs and Paley tournaments, demonstrating their concentration around the expected values in the corresponding random (di)graph models. In the context of coding theory, we establish a correspondence between even/odd induced sub(di)graphs of Paley graphs (tournaments) and maximum distance separable (MDS) self-dual codes that can be constructed via (extended) generalized Reed-Solomon codes from subsets of finite fields. As a consequence, our contribution on induced subgraphs leads to new existence and counting results about MDS self-dual codes.