An Erd\H{o}s-Gallai-type theorem for keyrings

A keyring is a graph obtained by appending $r\ge 1$ leaves to one of the vertices of a cycle. We prove that for every $r\le (k-1)\mathrm{/}2$, any graph with average degree more than $k-1$ contains a keyring with $r$ leaves and at least $k$ edges.