[Paper Review] Counting multiple graphs in generalized Turán problems

Dániel Gerbner|arXiv (Cornell University)|Jul 22, 2020

Limits and Structures in Graph Theory25 references4 citations

TL;DR

This paper introduces a generalized Turán problem that counts multiple subgraphs simultaneously in F-free graphs, proposing both uncolored and colored variants. It proves that for triangle-free graphs, the Turán graph T₂(n) maximizes the total count of specific subgraphs like matchings, cycles, and paths, establishing extremal results for generalized Turán and Berge hypergraph problems.

ABSTRACT

We are given graphs $H_1,\dots,H_k$ and $F$. Consider an $F$-free graph $G$ on $n$ vertices. What is the largest sum of the number of copies of $H_i$? The case $k=1$ has attracted a lot of attention. We also consider a colored variant, where the edges of $G$ are colored with $k$ colors. What is the largest sum of the number of copies of $H_i$ in color $i$? Our motivation to study this colored variant is a recent result stating that the Turán number of the $r$-uniform Berge-$F$ hypergraphs is at most the quantity defined above for $k=2$, $H_1=K_r$ and $H_2=K_2$. In addition to studying these new questions, we obtain new results for generalized Turán problems and also for Berge hypergraphs.

Motivation & Objective

• To extend generalized Turán problems by simultaneously counting multiple subgraphs H₁,…,Hₖ in F-free graphs, rather than just one.
• To introduce a colored variant where edges are colored and subgraph counts are aggregated per color, motivated by connections to Berge hypergraphs.
• To determine when the Turán graph T_{r}(n) maximizes the total number of copies of a family of subgraphs in F-free graphs.
• To establish extremal results for specific subgraphs such as matchings, C₅, P₅, K₂,₃, and C₄′ in triangle-free graphs.
• To prove that the Turán graph T₂(n) achieves the maximum sum of subgraph counts for certain families of graphs in the absence of triangles.

Proposed method

• Define the uncolored extremal function ex(n, (H₁,…,Hₖ), F) as the maximum sum of subgraph counts ∑ᵢ 𝒩(Hᵢ, G) over all F-free n-vertex graphs G.
• Introduce the colored variant ex^col(n, (H₁,…,Hₖ), F), where edges are colored and counts are aggregated per color, with Gᵢ being the subgraph of edges of color i.
• Use double counting and extremal graph theory techniques, particularly bounding edge sets in non-bipartite and bipartite subgraphs via Brouwer's theorem.
• Apply degree-based counting arguments: for a vertex v of degree d, bound the number of copies of subgraphs containing v by considering edge and vertex choices on the remaining n−1 or n−2 vertices.
• Compare counts in arbitrary F-free graphs G to those in the Turán graph T₂(n), showing that counts in G are at most those in T₂(n) via vertex-by-vertex extremality arguments.
• Leverage known results such as Brouwer’s bound on the number of edges in non-bipartite graphs and degree constraints in triangle-free graphs to refine upper bounds.

Experimental results

Research questions

• RQ1 When does the Turán graph T₂(n) maximize the total number of copies of multiple subgraphs H₁,…,Hₖ in triangle-free graphs?
• RQ2 How does the colored variant of the generalized Turán problem relate to Berge hypergraphs, particularly in the case k=2, H₁=K_r, H₂=K₂?
• RQ3 Can extremal counts for subgraphs like C₅, M (a 5-vertex matching), M′ (a 5-vertex matching with a triangle), P₅, and K₂,₃ be maximized by T₂(n) in triangle-free graphs?
• RQ4 What is the relationship between the uncolored and colored variants of the generalized Turán problem in terms of asymptotic growth and extremal graphs?
• RQ5 Under what conditions does the extremal graph for individual subgraphs Hᵢ also maximize the sum ∑ᵢ 𝒩(Hᵢ, G) in F-free graphs?

Key findings

• For the family of graphs T = {M, M′, C₄′, P₅, K₂,₃, C₅}, the Turán graph T₂(n) maximizes the total number of copies among all triangle-free n-vertex graphs, i.e., ex(n, T, K₃) = 𝒩(T; T₂(n)).
• The number of copies of M and C₅ in any triangle-free graph G is at most the number in T₂(n), as shown by vertex-by-vertex counting and comparison using degree and edge bounds.
• The number of copies of M′ and C₅ in any triangle-free graph G is also at most the number in T₂(n), with the proof relying on refined bounds when the remaining graph is non-bipartite or has low minimum degree.
• The colored variant ex^col(n, (H₁,…,Hₖ), F) is bounded between ex(n, Hᵢ, F) and ∑ᵢ ex(n, Hᵢ, F), and both variants are Θ(maxᵢ ex(n, Hᵢ, F)).
• When each Hᵢ is a clique or complete multipartite graph, the extremal graph for ∑ᵢ 𝒩(Hᵢ, G) in F-free graphs is a complete (ℓ−1)-partite graph, generalizing known results for single subgraphs.
• The result extends to families of subgraphs where each individual Hᵢ is maximized by T₂(n) in triangle-free graphs, implying the sum is also maximized by T₂(n).