[Paper Review] The shifting method and generalized Tur\'{a}n number of matchings

Jian Wang|arXiv (Cornell University)|Dec 5, 2018

Limits and Structures in Graph Theory11 references4 citations

TL;DR

This paper determines the exact generalized Turán number for matchings by applying the shifting method to compute the maximum number of copies of $K_s$ and $K^*_{s,t}$ in $M_{k+1}$-free graphs and bipartite graphs. It establishes tight upper bounds that match constructions from complete graphs and complete split graphs, resolving exact extremal values for these generalized Turán problems.

ABSTRACT

Given two graphs $T$ and $F$, the maximum number of copies of $T$ in an $F$-free graph on $n$ vertices is called the generalized Tur\\'{a}n number, denoted by $ex(n,T,F)$. When $T=K_2$, it reduces to the classical Tur\\'{a}n number $ex(n,F)$. Let $M_{k}$ be a matching with $k$ edges and $K^{*}_{s,t}$ a graph obtained from $K_{s,t}$ by replacing the part of size $s$ by a clique of the same size. In this paper, we show that for any $s\\geq 2$ and $n\\geq 2k+1$, \\[ ex(n,K_s,M_{k+1})=\\max\\left\\{\\binom{2k+1}{s}, \\binom{k}{s}+(n-k)\\binom{k}{s-1}\ ight\\}. \\] For any $s\\geq 1$, $t\\geq 2$ and $n\\geq 2k+1$, \\[ ex(n,K_{s,t}^*,M_{k+1})=\\max\\left\\{\\binom{2k+1}{s+t}\\binom{s+t}{t}, \\binom{k}{s}\\binom{n-s}{t}+(n-k)\\binom{k}{s+t-1}\\binom{s+t-1}{t}\ ight\\}. \\] Moreover, we also study the bipartite case of the problem. Let $ex_{bip}(n,T,F)$ be the maximum possible number of copies of $T$ in an $F$-free bipartite graph with each part of size $n$. We prove that for any $s,t\\geq 1$ and $n\\geq k$, \\[ ex_{bip}(n,K_{s,t},M_{k+1})=\\left\\{ \\begin{aligned} &\\binom{k}{s}\\binom{n}{t}+\\binom{k}{t}\\binom{n}{s}, & \\quad s\ eq t, &\\binom{k}{s}\\binom{n}{s},&\\quad s=t. \\end{aligned} \ ight. \\] Our proof is mainly based on the shifting method.

Motivation & Objective

• To determine the exact value of the generalized Turán number $ex(n, T, M_{k+1})$ for $T = K_s$ and $T = K^*_{s,t}$ in $M_{k+1}$-free graphs.
• To extend the generalized Turán problem to the bipartite setting, computing $ex_{ ext{bip}}(n, K_{s,t}, M_{k+1})$ for balanced bipartite graphs.
• To establish tight upper bounds that match extremal constructions, resolving open cases in generalized Turán theory.
• To demonstrate the effectiveness of the shifting method in preserving or increasing the number of clique and $K^*_{s,t}$ copies under edge redistribution.

Proposed method

• The shifting method is applied to transform any $M_{k+1}$-free graph into a shifted graph that preserves or increases the number of $K_s$ and $K^*_{s,t}$ copies.
• The shifted graph is shown to be a complete graph on $2k+1$ vertices or a complete split graph $K_k \vee E_{n-k}$, which are extremal configurations.
• For $K^*_{s,t}$, the proof classifies copies into three types based on vertex distribution across sets $U_0$, $U'$, and $U \setminus U_0$, and uses convexity of counting functions to bound the maximum.
• In the bipartite case, the König-Hall theorem is used to identify a vertex cover $T$ of size $k$, leading to a complete split bipartite graph $G^*$ that dominates all $M_{k+1}$-free graphs.
• The number of $K_{s,t}$ copies in $G^*$ is computed as a function of $|X_1| = x$, and convexity is used to find the maximum over $x \in \{0, \dots, k\}$.
• The extremal values are derived by evaluating the maximum of piecewise functions at endpoints $x=0$ and $x=k$, yielding closed-form expressions.

Experimental results

Research questions

• RQ1 What is the maximum number of $K_s$ copies in an $M_{k+1}$-free graph on $n$ vertices for $s \geq 2$ and $n \geq 2k+1$?
• RQ2 What is the maximum number of $K^*_{s,t}$ copies in an $M_{k+1}$-free graph on $n$ vertices for $s \geq 1$, $t \geq 2$, and $n \geq 2k+1$?
• RQ3 What is the maximum number of $K_{s,t}$ copies in an $M_{k+1}$-free balanced bipartite graph with parts of size $n$?
• RQ4 Can the shifting method be used to preserve or increase the number of $K_s$ and $K^*_{s,t}$ copies while avoiding $M_{k+1}$?
• RQ5 What are the extremal graphs achieving the maximum number of $K_s$ and $K^*_{s,t}$ copies in $M_{k+1}$-free graphs?

Key findings

• For $s \geq 2$ and $n \geq 2k+1$, the generalized Turán number satisfies $ex(n, K_s, M_{k+1}) = \max\left\{\binom{2k+1}{s}, \binom{k}{s} + (n-k)\binom{k}{s-1}\right\}$.
• For $s \geq 1$, $t \geq 2$, and $n \geq 2k+1$, $ex(n, K^*_{s,t}, M_{k+1}) = \max\left\{\binom{2k+1}{s+t}\binom{s+t}{t}, \binom{k}{s}\binom{n-s}{t} + (n-k)\binom{k}{s+t-1}\binom{s+t-1}{t}\right\}$.
• In the bipartite case, for $s \neq t$, $ex_{\text{bip}}(n, K_{s,t}, M_{k+1}) = \binom{k}{s}\binom{n}{t} + \binom{k}{t}\binom{n}{s}$, and for $s = t$, it is $\binom{k}{s}\binom{n}{s}$.
• The extremal graphs achieving these bounds are the complete graph $K_{2k+1}$ and the complete split graph $K_k \vee E_{n-k}$ for the non-bipartite case.
• The shifting operation preserves or increases the number of $K_s$ and $K^*_{s,t}$ copies, and the resulting extremal graphs are shown to be optimal via convexity of counting functions.
• The proof relies on convexity of the functions counting $K_{s,t}$ copies in the shifted graph, ensuring the maximum occurs at the endpoints $\ell = 2k+1$ or $\ell = k+1$.