The Erdos-Sos Conjecture For Spiders Of Four Legs

A classical result on extremal graph theory is the Erdos-Gallai theorem:if G is a graph of order n and size e(G)>(k-1)n/2, then G contains a path of length k This motivated Erdos and Sos to propose the following conjecture in1963:if G is a graph of order n and size e(G)>(k-1)n/2, then G contains every tree of size kThe Erdos-Sos conjecture is still open. However, several special cases of the conjecture have been confirmed, which will be listed in the section of introduction. In2007, Fan and Sun proved the conjecture is true for spiders of three legs. Making good use of the extremal properties, we prove in this thesis that the conjecture is true for spiders of four legs, one of which has length at least k/2.