On the Erdés-Sés Conjecture and double-brooms

Let \( G \) be a graph with average degree greater than \( k – 2 \). Erdős and Sós conjectured that \( G \) contains every tree on \( k \) vertices. A star is a tree consisting of one center vertex adjacent to all the other vertices, and a \emph{double-broom} is a tree made up of two stars and a path connecting the center of one star with the center of the other. If the path connecting the two stars has length 2 or 3, then \( G \) contains the double-broom (unpublished). In this paper, we prove that \( G \) contains every double-broom on \( k \) vertices.