[Paper Review] Generalized Tur\'an problems for disjoint copies of graphs

Dániel Gerbner, Abhishek Methuku+1 more|arXiv (Cornell University)|Dec 19, 2017

Limits and Structures in Graph Theory9 references3 citations

TL;DR

This paper investigates generalized Turán numbers for disjoint copies of graphs, specifically $ ex(n,H,kF) $, the maximum number of copies of a graph $ H $ in an $ n $-vertex graph avoiding $ k $ vertex-disjoint copies of a fixed graph $ F $. It establishes asymptotic bounds for $ H = K_3 $ and various $ F $, showing that $ ex(n,K_3,kC_5) = (1+o(1))inom{k-1}{3} rac{n^6}{4^3} $, revealing a significant jump in order of magnitude compared to $ ex(n,K_3,C_5) $, and introduces a novel counting method using universal vertices and extremal graph structures.

ABSTRACT

Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $ex(n,H,F)$. We investigate the function $ex(n,H,kF)$, where $kF$ denotes $k$ vertex disjoint copies of a fixed graph $F$. Our results include cases when $F$ is a complete graph, cycle or a complete bipartite graph.

Motivation & Objective

• To understand the behavior of $ ex(n,H,kF) $, the maximum number of copies of $ H $ in an $ n $-vertex $ kF $-free graph, where $ kF $ denotes $ k $ vertex-disjoint copies of $ F $.
• To investigate how the extremal count of copies of $ H $ changes when forbidding $ k $ disjoint copies of $ F $, compared to forbidding just one copy.
• To establish asymptotic or order-of-magnitude bounds for $ ex(n,H,kF) $ when $ H $ is a triangle and $ F $ is a cycle, complete graph, or complete bipartite graph.
• To explore structural properties of extremal graphs that maximize the number of $ H $-copies under $ kF $-freeness, particularly using universal vertices and Turán-type constructions.
• To identify cases where $ ex(n,H,kF) $ grows significantly faster than $ ex(n,H,F) $, challenging the intuition from classical Turán theory.

Proposed method

• Uses the construction $ K_{k-1} + G $, where $ G $ is $ F $-free, to generate $ kF $-free graphs with many copies of $ H $, leveraging the $ k-1 $ universal vertices to block $ k $ disjoint copies of $ F $.
• Applies extremal graph theory techniques, particularly the use of Turán graphs $ T_r(n) $, to bound the number of $ H $-copies in $ F $-free graphs.
• Employs a 'good copy' counting method for $ lK_3 $, where each triangle in the copy uses at most one vertex from each of $ r riangleq k-1 $ fixed triangles in the graph.
• Uses double counting and edge-sharing arguments to bound the number of good triangles containing vertices from a fixed triangle $ A_i $, showing it is $ O(n) $ per triangle.
• Reduces the problem of counting $ lK_3 $ to counting $ lK_2 $ in a triangle-free graph $ G_R $, using the bound $ ex(n,lK_2,K_3) = (1+o(1)) rac{1}{l!} inom{n^2}{4} $ from Erdős’s result.
• Applies the probabilistic and extremal counting framework to derive asymptotic expressions for $ ex(n,K_3,kF) $, especially for $ F = C_5 $, by combining universal vertex contributions and independent edge pairings.

Experimental results

Research questions

• RQ1 How does $ ex(n,K_3,kC_5) $ scale asymptotically as $ n $ grows, and how does it compare to $ ex(n,K_3,C_5) $?
• RQ2 What structural properties of $ kF $-free graphs maximize the number of copies of $ H $, particularly when $ H = K_3 $ and $ F $ is a cycle or complete graph?
• RQ3 Can the order of magnitude of $ ex(n,H,kF) $ be significantly larger than $ ex(n,H,F) $, and if so, under what conditions?
• RQ4 What is the asymptotic behavior of $ ex(n,K_3,kF) $ when $ F $ is a complete graph $ K_r $ or a complete bipartite graph $ K_{s,t} $?
• RQ5 To what extent can the method of universal vertices and extremal graph decomposition be generalized to other graphs $ H $ and $ F $?

Key findings

• For $ F = C_5 $, $ ex(n,K_3,kC_5) = (1+o(1))inom{k-1}{3} rac{n^6}{64} $, showing a polynomial jump in growth rate compared to $ ex(n,K_3,C_5) = heta(n^{3/2}) $.
• The number of good copies of $ lK_3 $ in a $ kC_5 $-free graph is at most $ (1+o(1))inom{k-1}{l} rac{n^{2l}}{4^l} $, derived by counting $ lK_2 $ in a triangle-free graph and pairing with $ l $ vertices from $ k-1 $ fixed triangles.
• The contribution of triangles using universal vertices is bounded by $ O(n) $ per fixed triangle, ensuring that such configurations do not dominate the count beyond lower-order terms.
• The construction $ K_{k-1} + T_2(n-k+1) $, where $ T_2(n-k+1) $ is the complete bipartite Turán graph, achieves $ heta(n^6) $ triangles when $ F = C_5 $, demonstrating the tightness of the upper bound.
• For $ F = C_{2l+1} $ or $ C_{2l} $, $ ex(n,K_r,C_k) = O(n^{1+1/l}) $, extending known results on Berge hypergraphs to the generalized Turán setting.
• The paper shows that $ ex(n,H,kF) $ can grow much faster than $ ex(n,H,F) $, as demonstrated by the $ heta(n^6) $ vs. $ heta(n^{3/2}) $ contrast for $ H = K_3 $, $ F = C_5 $, and $ k=2 $, indicating a fundamental difference from classical Turán numbers.