An Erd\H{o}s-Gallai-type theorem for keyrings with larger number of leaves

TL;DR

This paper extends an Erd ext{"o}s-Gallai-type theorem to larger leaf counts in keyrings, providing a more comprehensive understanding of the conditions under which such structures exist in graphs.

Contribution

The authors prove that Sidorenko's theorem for keyrings applies to larger numbers of leaves, completing the Erd ext{"o}s-Gallai-type characterization for keyrings.

Findings

Sidorenko's theorem holds for larger leaf counts in keyrings.

The result completes the Erd ext{"o}s-Gallai-type theorem for keyrings.

The theorem applies to graphs with certain size and order conditions.

Abstract

A keyring is a graph obtained from a cycle by appending $r\ge0$ leaves to one of its vertices. Sidorenko proved an Erd\H{o}s-Gallai-type theorem: Every graph of order $n$ and size more than $\frac{(k-1)n}{2}$ contains a keyring of size at least $k$ and with $r$ leaves for $r\le\frac{k-1}2$ (Theorem 1.4, An Erd\H{o}s-Gallai-type theorem for keyrings, Graphs Combin., 2018). In this note, we show that Sidorenko's theorem holds for larger $r$ and so complete the Erd\H{o}s-Gallai-type theorem for keyrings.