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

Qilong Li, Yue Zhou

TL;DR

The paper studies induced subgraphs and subdigraphs of Paley graphs and Paley tournaments under degree-parity constraints, developing both exact counts and asymptotics. It leverages Gallai-type parity partitions, parity-cover dualities via odd-extensions, and linear-algebra/number-theoretic tools (Weil sums, Smith normal form) to obtain results that distinguish Paley graphs by $q\bmod 8$ and to show nonexistence for Paley tournaments. A central contribution is the precise enumeration of co-even induced subgraphs in Paley graphs and the asymptotic counts for small-order even induced subgraphs, together with a robust coding-theory bridge to MDS self-dual codes constructed from (extended) GRS codes. This connection yields new existence and counting results for MDS self-dual codes and advances understanding of parity-constrained subgraph structures in Paley graphs and tournaments, with implications for explicit code constructions.

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.

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

TL;DR

The paper studies induced subgraphs and subdigraphs of Paley graphs and Paley tournaments under degree-parity constraints, developing both exact counts and asymptotics. It leverages Gallai-type parity partitions, parity-cover dualities via odd-extensions, and linear-algebra/number-theoretic tools (Weil sums, Smith normal form) to obtain results that distinguish Paley graphs by and to show nonexistence for Paley tournaments. A central contribution is the precise enumeration of co-even induced subgraphs in Paley graphs and the asymptotic counts for small-order even induced subgraphs, together with a robust coding-theory bridge to MDS self-dual codes constructed from (extended) GRS codes. This connection yields new existence and counting results for MDS self-dual codes and advances understanding of parity-constrained subgraph structures in Paley graphs and tournaments, with implications for explicit code constructions.

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 and is trivial if , 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 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.
Paper Structure (14 sections, 24 theorems, 94 equations, 1 table)

This paper contains 14 sections, 24 theorems, 94 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Even-even partition and induced subgraphs with degree parity conditions
  4. Motivation from coding theory and related problems
  5. Binomial coefficients
  6. Giant even induced subgraphs of undirected graphs
  7. Co-even induced subgraphs of Paley graphs
  8. Characterization of co-even induced subgraphs
  9. Paley graphs
  10. The number of even induced subgraphs of order $o(q^{1/4})$ in Paley graphs and Paley tournaments
  11. Paley graphs
  12. Paley tournaments
  13. Applications to the construction of MDS self-dual codes
  14. Expectations in random (di)graphs

Key Result

Theorem 1.1

Let $r\geq 3$ and let $q\equiv 1\ (\mathrm{mod}\ 4)$ be a prime power satisfying Then $P_{q}$ contains every graph of order $r$ as an induced subgraph.

Theorems & Definitions (40)

  • Theorem 1.1: bollobas2001random
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: Gallai's Theorem
  • Theorem 2.2: caro1996simple
  • Lemma 2.3
  • Theorem 2.4
  • Lemma 2.5
  • ...and 30 more