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

TL;DR

This paper studies the parity conditions of degrees in induced subgraphs of Paley graphs and tournaments, providing bounds, formulas, and connections to MDS self-dual codes, advancing understanding in graph theory and coding theory.

Contribution

It introduces new bounds, formulas, and existence results for parity-constrained induced subgraphs in Paley graphs and tournaments, linking them to MDS self-dual codes.

Findings

Lower bounds for large even induced subgraphs in Paley graphs.

Exponential count of even-even partitions when q ≡ 1 mod 8.

Asymptotic formulas for even induced subgraphs, showing concentration around expected values.

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.