Advances on the Conjecture of Erdős-Sós for spiders

Abstract

- A hamiltonian graph $G$ verifying $e(G)>n(k-1)/2$ %with a vertex of degree greater or equal than $k$ contains any $k$-spider. - If $G$ is a graph with average degree $\bar{d}>k-1$, then every spider of size $k$ is contained in $G$ for $k\le 10$. - A $2$-connected graph with average degree $\bar{d}>{\ell }_2+{\ell }_3+{\ell }_4$ contains every spider of $4$ legs $S_{1,{\ell }_2,{\ell }_3,{\ell }_4}$. We claim also that the condition of $2$-connection is not needed, but the proof is very long and it is not included in this document.

6 pages, 2*3 figures