The strong (non-induced) Turán numbers

Source abstract

In this paper we introduce and explore the following new graph invariant: For a graph $G$ on $k$ vertices, $G \neq K_k$, let $st(n,G)$ denote the maximum number of edges in a graph of order $n$ which does not contain any subgraph on $k$ vertices strictly containing $G$. A basic relation to classical Turán numbers is developed via the following: For $G$ on $k$ vertices, let $D(G) = \{ H : |H| = |G|, H = G + e \}$. Using this notion we prove that $ex(n,G) \leq st(n,G) = ex(n, D(G) ) \leq \min \{ ex(n,H) : H \in D(G) \}$ holds for all $n \geq |G|$. The family $D(G)$ happened to be smoothly amenable to the use of classical extremal results, and in many cases allows us to get asymptotically sharp estimates as well as exact values of $st(n,G)$. From the many results proved here we state the following as an illustration. (1) If $χ(G) \geq 3$ and $χ(D(G)) = χ(G)$, then $st(n,G) = (1+o(1))ex(n,K_{χ(G)})$. (2) If $χ(D(G)) = χ(G) +1$, then $G$ is a complete $χ(G)$-partite graph and $st(n,G) = ex(n,K_{χ(G) +1})$ for $n$ sufficiently large. (3) For $k$ odd, $k\geq 5$, $st(n,C_k) = ex(n,C_k) = ex(n,K_3)$ for $n$ sufficiently large. (4) If $T$ is a tree of order $q$ with diameter $k \geq 2$ and $q \geq k+1 \geq 3$, then $ex(n, \{C_3,...,C_{k+1}\}) \leq st(n,T) \leq ex(n, \{C_3,...,C_{k+1}\}) + (q-1)n$. Many results concerning even cycles, theta graphs, dense bipartite graphs and graphs of the form $G = G^* \cup tK_1$ are obtained, moreover the value of $st(n,G)$ is computed for all graphs on at most 4 vertices.