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{\bf Diana Piguet and Maya Jakobine Stein}
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{\bf The Loebl--Koml\'os--S\'os Conjecture for Trees of Diameter $5$ and for Certain Caterpillars}
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Loebl, Koml\'os, and S\'os conjectured that if at least half the
vertices of a graph $G$ have degree at least some $k\in \Bbb N$, then
every tree with at most~$k$ edges is a subgraph of $G$.
We prove the conjecture for all trees of diameter at most~$5$ and for
a class of caterpillars. Our result implies a bound on the Ramsey
number $r(T,T')$ of trees~$T,T'$ from the above classes.
\bye