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arXiv · 1606.09343

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

Frédéric Havet·Bruce Reed·Maya Stein·David R. Wood
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Abstract

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

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Frédéric Havet, Bruce Reed, Maya Stein, David R. Wood. 2016-06-30. A Variant of the Erd\H{o}s-S\'os Conjecture. https://doi.org/10.1002/jgt.22511

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