Constructing Trees in Graphs whose Complement has no K_2,s

Abstract

We show that, if G is a graph of order n with maximal degree Δ(G) and minimal degree δ(G) whose complement contains no K_2,s, s [ges ] 2, then G contains every tree T of order n−s+1 whose maximal degree is at most Δ(G) and whose vertex of second-largest degree is at most δ(G). We then show that this result implies that special cases of two conjectures are true. We verify that the Erdös–Sós conjecture, which states that a graph whose average degree is larger than k−1 contains every tree of order k+1, is true for graphs whose complement does not contain a K_2,4, and the Komlós–Sós conjecture, which states that every graph of median degree at least k contains every tree of order k+1, is true for graphs whose complement does not contain a K_2,3.