[Paper Review] Some sharp results on the generalized Tur\'an numbers

Jie Ma, Yu Qiu|arXiv (Cornell University)|Feb 4, 2018

Limits and Structures in Graph Theory11 references8 citations

TL;DR

This paper provides sharp estimates for the generalized Turán number ex(n, T, H), focusing on the maximum number of copies of a graph T in an H-free n-vertex graph when χ(T) < χ(H). It establishes that ex(n, K_m, H) = N(T_r(n), K_m) + biex(n, H) · Θ(n^{m−2}) by introducing the decomposition family of H and the biex(n, H) function, refining the error term in Alon and Shikhelman's o(n^m) bound. The result extends Erdős' classical theorem and proves uniqueness of the Turán graph T_r(n) for edge-critical H.

ABSTRACT

For graphs $T, H$, let $ex(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex $H$-free graph. In this paper we prove some sharp results on this generalization of Tur\\'an numbers, where our focus is for the graphs $T,H$ satisfying $\\chi(T)&lt;\\chi(H)$. This can be dated back to Erd\\H{o}s, where he generalized the celebrated Tur\\'an's theorem by showing that for any $r\\geq m$, the Tur\\'an graph $T_r(n)$ uniquely attains $ex(n,K_m,K_{r+1})$. For general graphs $H$ with $\\chi(H)=r+1&gt;m$, Alon and Shikhelman showed that $ex(n,K_m,H)=\\binom{r}{m}(\\frac{n}{r})^m+o(n^m)$. Here we determine this error term $o(n^m)$ up to a constant factor. We prove that $ex(n,K_m,H)=\\binom{r}{m}(\\frac{n}{r})^m+biex(n,H)\\cdot\\Theta(n^{m-2})$, where $biex(n,H)$ is the Tur\\'an number of the decomposition family of $H$. As a special case, we extend Erd\\H{o}s' result, by showing that $T_r(n)$ uniquely attains $ex(n,K_m,H)$ for any edge-critical graph $H$. We also consider $T$ being non-clique, where even the simplest case seems to be intricate. Following from a more general result, we show that for all $s\\leq t$, $T_2(n)$ maximizes the number of $K_{s,t}$ in $n$-vertex triangle-free graphs if and only if $t&lt;s+\\frac12+\\sqrt{2s+\\frac14}$.

Motivation & Objective

• To refine the error term o(n^m) in Alon and Shikhelman's generalized Turán number ex(n, K_m, H) to a tight Θ(n^{m−2}) factor.
• To determine the exact asymptotic behavior of ex(n, K_m, H) when χ(H) = r+1 > m, by introducing the decomposition family of H and the biex(n, H) function.
• To prove that the Turán graph T_r(n) uniquely maximizes the number of K_m copies in H-free graphs when H is edge-critical.
• To extend the analysis beyond cliques T to complete multipartite graphs T, particularly K_{s,t}, in triangle-free graphs.
• To investigate the extremal structure of graphs maximizing T-copies in K_{r+1}-free graphs, especially when T is not a clique.

Proposed method

• Define the decomposition family F_H of H as the set of all bipartite graphs formed by deleting r−1 color classes from an (r+1)-coloring of H.
• Introduce biex(n, H) as the maximum number of edges in an n-vertex graph avoiding all graphs in F_H as subgraphs.
• Establish a stability result (Theorem 1.4): if G is H-free and contains nearly as many K_m copies as T_r(n), then G is a O(n²)-edge perturbation of T_r(n).
• Construct a lower bound by inserting an F_H-free graph F with biex(n, H) edges into the largest part of T_r(n), ensuring G remains H-free and gains Ω(biex(n, H) · n^{m−2}) copies of K_m.
• Use the Kövári–Sós–Turán theorem to bound biex(n, H) = O(n^{2−α_H}) for some α_H > 0, improving the error term to O(n^{m−α_H}).
• Apply extremal graph theory and stability techniques to analyze the structure of extremal graphs for complete multipartite T = K_{s,t} in triangle-free graphs.

Experimental results

Research questions

• RQ1 What is the exact asymptotic order of the error term in ex(n, K_m, H) beyond o(n^m), particularly when χ(H) = r+1 > m?
• RQ2 For which graphs H is the Turán graph T_r(n) the unique maximizer of K_m copies in H-free graphs?
• RQ3 How does the structure of extremal graphs change when T is not a clique, such as K_{s,t} in triangle-free graphs?
• RQ4 Under what conditions on s and t is the complete bipartite graph K_{s,t} maximized in T_2(n) among all triangle-free n-vertex graphs?
• RQ5 Can the extremal graph for ex(n, T, K_{r+1}) be characterized when T is a complete multipartite graph or an odd cycle?

Key findings

• The error term in ex(n, K_m, H) is precisely Θ(biex(n, H) · n^{m−2}), where biex(n, H) is the extremal number for the decomposition family F_H of H.
• For edge-critical H with χ(H) = r+1 > m, biex(n, H) = 0, so ex(n, K_m, H) = N(T_r(n), K_m), and T_r(n) is the unique extremal graph.
• The result generalizes Erdős’s theorem: T_r(n) uniquely maximizes K_m copies in K_{r+1}-free graphs, even when H is any edge-critical graph with χ(H) = r+1.
• For K_{s,t}-maximization in triangle-free graphs, T_2(n) is optimal if and only if t < s + 1/2 + √(2s + 1/4), a sharp threshold condition.
• The stability result (Theorem 1.4) shows that any H-free graph with nearly maximal K_m count must be a small perturbation (o(n²) edges) of T_r(n).
• The analysis reveals that extremal graphs for T = K_{s,t} in triangle-free settings are not always bipartite when t ≥ s + 1/2 + √(2s + 1/4), suggesting structural complexity beyond bipartite extremals.