Let F be a finite family of graphs with ₅ ₅χ (F) =r+13, where χ (F) is the chromatic number of F. Set t=₅|F|. Let EX (n, F) be the set of graphs with maximum edges among all the graphs of order n without any F as a subgraph. Let T (n, r) be the Turán graph of order n with r parts. Assume that some F₀ is a subgraph of the graph obtained from T (rt, r) by embedding a path in its one part. Simonovits S1 introduced the concept of symmetric subgraphs, and proved that there exist graphs in EX (n, F) which have symmetrical property. In this paper, we aim to find a way to characterize all the extremal graphs for such F using symmetric subgraphs. Some new extremal results are obtained.
Wenqian Zhang (Tue,) studied this question.