Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subgraphs with a positive minimum semidegree in digraphs with large outdegree

Andrzej Grzesik, Vojtech Rodl, Jan Volec

TL;DR

The paper establishes a sharp lower bound on the minimum semidegree induced in a subgraph of an n-vertex digraph with a given outdegree d, showing δ^±(H) ≥ d(d+1)/(2n). The proof uses a greedy vertex-removal strategy and a careful double-counting argument to derive the bound, which is shown to be asymptotically tight for d = o(n) via explicit tournament constructions. It also explores dense-digraph regimes, deriving sharper bounds for large α in δ^+(G) = αn, and extends the discussion to oriented graphs and normalized semidegree μ(G), connecting these bounds to cycles and regularity-based structures such as C_ℓ[b].

Abstract

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $δ^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{δ^+(H), δ^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$ then this bound is asymptotically best possible.

Subgraphs with a positive minimum semidegree in digraphs with large outdegree

TL;DR

The paper establishes a sharp lower bound on the minimum semidegree induced in a subgraph of an n-vertex digraph with a given outdegree d, showing δ^±(H) ≥ d(d+1)/(2n). The proof uses a greedy vertex-removal strategy and a careful double-counting argument to derive the bound, which is shown to be asymptotically tight for d = o(n) via explicit tournament constructions. It also explores dense-digraph regimes, deriving sharper bounds for large α in δ^+(G) = αn, and extends the discussion to oriented graphs and normalized semidegree μ(G), connecting these bounds to cycles and regularity-based structures such as C_ℓ[b].

Abstract

We prove that every -vertex directed graph with the minimum outdegree contains a subgraph satisfying We also show that if then this bound is asymptotically best possible.
Paper Structure (9 sections, 2 theorems, 24 equations, 1 figure)

This paper contains 9 sections, 2 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Our results
  4. Proof of Theorem \ref{['thm:main']}
  5. Tournaments with all subgraphs having small semidegree
  6. Dense digraphs
  7. Very dense digraphs
  8. Orientations of very dense undirected graphs
  9. Normalized minimum semidegree

Key Result

Theorem 1

Every loopless digraph $G$ with $n$ vertices and outdegree $\delta^+(G) \ge d$ has a subgraph $H$ with $\delta^\pm(H) \ge \frac{d(d+1)}{2n}$.

Figures (1)

  • Figure 1: The tournament constructed in Proposition \ref{['prop:construction']}.

Theorems & Definitions (3)

  • Theorem 1
  • Proposition 2
  • proof