Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. ,  study the problem of bounding t(G) for graphs which do not contain a complete graph K"r on r vertices. This problem was posed tnty years ago by Erdos, Saks, and Sos. Substantially improving earlier results of various researchers,  prove that every connected triangle-free graph on n vertices contains an induced tree of order n. When r>=4,  also show that t(G)>=logn4logr for every connected K"r-free graph G of order n. Both of these bounds are tight up to small multiplicative constants, and the first one disproves a recent conjecture of Matousek and Samal.
['Jacob Fox', 'Po-Shen Loh', 'Benny Sudakov']
I am writing a research paper on a idea Large induced trees in Kr -free graphs
[{"from": "human", "value": "I am writing a research paper on a idea Large induced trees in Kr -free graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. ,  study the problem of bounding t(G) for graphs which do not contain a complete graph K\"r on r vertices. This problem was posed tnty years ago by Erdos, Saks, and Sos. Substantially improving earlier results of various researchers,  prove that every connected triangle-free graph on n vertices contains an induced tree of order n. When r>=4,  also show that t(G)>=logn4logr for every connected K\"r-free graph G of order n. Both of these bounds are tight up to small multiplicative constants, and the first one disproves a recent conjecture of Matousek and Samal."}]