Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A classical result on extremal graph theory is the Erdos-Gallai theorem: if a graph on n vertices has more than (k-1)n2 edges, then it contains a path of k edges. Motivated by the result, Erdos and Sos conjectured that under the same condition, the graph should contain every tree of k edges. A spider is a rooted tree in which each vertex has degree one or two, except for the root. A leg of a spider is a path from the root to a vertex of degree one. Thus, a path is a spider of 1 or 2 legs. From the motivation, it is natural to consider spiders of 3 legs. ,  prove that if a graph on n vertices has more than (k-1)n2 edges, then it contains every k-edge spider of 3 legs, and also, every k-edge spider with no leg of length more than 4, which strengthens a result of Wozniak on spiders of diameter at most 4.
['Genghua Fan', 'Lingli Sun']
I am writing a research paper on a idea The Erdös-Sós conjecture for spiders
[{"from": "human", "value": "I am writing a research paper on a idea The Erd\u00f6s-S\u00f3s conjecture for spiders"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A classical result on extremal graph theory is the Erdos-Gallai theorem: if a graph on n vertices has more than (k-1)n2 edges, then it contains a path of k edges. Motivated by the result, Erdos and Sos conjectured that under the same condition, the graph should contain every tree of k edges. A spider is a rooted tree in which each vertex has degree one or two, except for the root. A leg of a spider is a path from the root to a vertex of degree one. Thus, a path is a spider of 1 or 2 legs. From the motivation, it is natural to consider spiders of 3 legs. ,  prove that if a graph on n vertices has more than (k-1)n2 edges, then it contains every k-edge spider of 3 legs, and also, every k-edge spider with no leg of length more than 4, which strengthens a result of Wozniak on spiders of diameter at most 4."}]