Minimum degree conditions for containing an $r$-regular $r$-connected subgraph

TL;DR

This paper determines the minimum degree threshold for large graphs to contain an $r$-regular, $r$-connected subgraph, resolving a question posed by Kriesell.

Contribution

It establishes the exact minimum degree condition for the existence of such subgraphs when $r$ is fixed and $n$ is large, answering an open problem.

Findings

Minimum degree condition: $rac{n+r-2}{2}$ for $r$-regular, $r$-connected subgraphs

Condition holds when $nr$ is even and $n$ is large

Provides a precise threshold for large graphs with fixed $r$

Abstract

We study optimal minimum degree conditions when an $n$-vertex graph $G$ contains an $r$-regular $r$-connected subgraph. We prove for $r$ fixed and $n$ large the condition to be $\delta(G) \ge \frac{n+r-2}{2}$ when $nr \equiv 0 \pmod 2$. This answers a question of M.~Kriesell.