The Erdos-Sos conjecture for spiders of four legs

The Erdos-Sos Conjecture states that if G is a graph with average degree more than k - 1, then G contains every tree of k edges. A special case of the conjecture is the well-known Erdos-Gallai theorem: if G is a graph with average degree more than k - 1, then G contains a path of k edges. A spider is a tree with at most one vertex of degree more than 2, called the center of the spider (if no vertex of degree more than two, then any vertex can be the center). A leg of a spider is a path from the center to a vertex of degree 1. Thus, a path can be regarded as a spider of 1 or 2 legs. In this paper, we prove that if G is a graph with average degree more than k - 1, then G contains every spider of 4 legs.