Terminal-Pairability in Complete Graphs
  We investigate terminal-pairability properties of complete graphs and improve the known bounds in two open problems. We prove that the complete graph $K_n$ on $n$ vertices is terminal-pairable if the maximum degree $\Delta$ of the corresponding demand graph $D$ is at most $2\lfloor\frac{n}{6}\rfloor-2$. We also verify the terminal-pairability property when the number of edges in $D$ does not exceed $2n-5$ and $\Delta\leq n-1$ holds. 
Ervin Gy\"ori, Tam\'as R\'obert Mezei, G\'abor M\'esz\'aros
math.CO