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A Variant of the Erd\H{o}s-S\'os Conjecture

Abstract

A well-known conjecture of Erd\H{o}s and S\'os states that every graph with average degree exceeding m1m-1 contains every tree with mm edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding mm and minimum degree at least 2m3\lfloor \frac{2m}{3}\rfloor contains every tree with mm edges. As evidence for our conjecture we show (i) for every mm there is a g(m)g(m) such that the weakening of the conjecture obtained by replacing mm by g(m)g(m) holds, and (ii) there is a γ>0\gamma>0 such that the weakening of the conjecture obtained by replacing 2m3\lfloor \frac{2m}{3}\rfloor by (1γ)m(1-\gamma)m holds

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