On Turán-good graphs

Some exact results for generalized Turán problems

“…It was shown in [14] that C 4 is F 2 -Turán-good, i.e., ex(n, C 4 , F 2 ) = max{N (H, T )}, while biex(n, F ) = 1. A more general result is in [10], showing that for any F with a color-critical vertex there is a graph H that is F -Turán-good. Again, the error term is 0 instead of biex(n, F )Θ(n |V (H)|−2 ).…”

On weakly Turán-good graphs

“…We remark that the first result concerning F -Turán-good graphs when F does not have a color-critical edge is due to Gerbner and Palmer [13], who showed that C 4 is F 2 -Turán-good, where F 2 consists of two triangles sharing a vertex. Gerbner [9] constructed F -Turán-good graphs for every F with a color-critical vertex, but they were always complete (χ(F ) − 1)partite graphs. In particular, K m,m = P 2 (m) is F -Turán-good.…”

“…A color-critical vertex is a vertex whose removal decreases the chromatic number. It was shown in [9] that for a given graph F , there exists an F -Turán-good graph if and only if F has a color-critical vertex. One can easily see from the proof in [9] that the same holds in the weak setting, i.e., there exists a weakly F -Turán-good graph if and only if F has a color-critical vertex.…”

Generalized Turán results for intersecting cliques

“…Gerbner and Palmer [20] and Gerbner [15] showed that we also have ex(n, P 3 , F ) = N (P 3 , T (n, r − 1)). Gerbner [16] presented a theorem that determines the exact value of ex(n, H, B r,s ) for a class of graphs H if n is large enough.…”

“…a vertex whose deletion decreases the chromatic number (from r to r−1). Gerbner [16] determined ex(n, H, F ) for every r-chromatic graph F with a color-critical vertex if H is a complete balanced (r − 1)-partite graph K a,...,a with a large enough. In particular, this determines ex(n, K a,a , B 3,1 ) for every a.…”