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Localization: A Framework to Generalize Extremal Graph Problems

Rajat Adak, L. Sunil Chandran

TL;DR

The localization framework is employed to improve upper bounds on the number of $K_t$-cliques in graphs of bounded maximum degree and provide structural characterizations of the extremal graphs attaining them.

Abstract

Extremal graph theory studies the maximum or minimum number of subgraphs isomorphic to a prescribed graph under given constraints. \textit{Localization} has recently emerged as a framework that refines such problems by assigning extremal quantities locally (to vertices or edges) and then aggregating them. This perspective not only recovers classical results but also leads to sharper bounds. A classical result states that a connected planar graph with a finite girth $g$ satisfies \begin{equation*} m \leq \frac{g}{g-2}(n-2) \end{equation*} Wood~\cite{wood} derived upper bounds on the number of $K_t$-cliques in graphs of bounded maximum degree, expressed in terms of both the number of vertices and the number of edges: \begin{align*} ex(n,K_t,K_{1,d+1}) \leq \frac{n}{d+1}\binom{d+1}{t} \\ mex(m,K_t,K_{1,d+1}) \leq \frac{m}{\binom{d+1}{2}}\binom{d+1}{t} \end{align*} More recently, Chakraborty and Chen~\cite{CHAKRABORTI2024103955} established a similar upper bound for graphs with bounded path length: \begin{equation*} mex(m,K_t,P_{r+1}) \leq \frac{m}{\binom{r}{2}}\binom{r}{t} \end{equation*} In this paper, we employ the localization framework to improve these bounds and provide structural characterizations of the extremal graphs attaining them.

Localization: A Framework to Generalize Extremal Graph Problems

TL;DR

The localization framework is employed to improve upper bounds on the number of -cliques in graphs of bounded maximum degree and provide structural characterizations of the extremal graphs attaining them.

Abstract

Extremal graph theory studies the maximum or minimum number of subgraphs isomorphic to a prescribed graph under given constraints. \textit{Localization} has recently emerged as a framework that refines such problems by assigning extremal quantities locally (to vertices or edges) and then aggregating them. This perspective not only recovers classical results but also leads to sharper bounds. A classical result states that a connected planar graph with a finite girth satisfies \begin{equation*} m \leq \frac{g}{g-2}(n-2) \end{equation*} Wood~\cite{wood} derived upper bounds on the number of -cliques in graphs of bounded maximum degree, expressed in terms of both the number of vertices and the number of edges: \begin{align*} ex(n,K_t,K_{1,d+1}) \leq \frac{n}{d+1}\binom{d+1}{t} \\ mex(m,K_t,K_{1,d+1}) \leq \frac{m}{\binom{d+1}{2}}\binom{d+1}{t} \end{align*} More recently, Chakraborty and Chen~\cite{CHAKRABORTI2024103955} established a similar upper bound for graphs with bounded path length: \begin{equation*} mex(m,K_t,P_{r+1}) \leq \frac{m}{\binom{r}{2}}\binom{r}{t} \end{equation*} In this paper, we employ the localization framework to improve these bounds and provide structural characterizations of the extremal graphs attaining them.

Paper Structure

This paper contains 11 sections, 17 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Localization
  3. Main Results
  4. Recovering the classical bounds
  5. Proof of Results
  6. Proof of \ref{['girthlocal']}
  7. Proof of \ref{['thm2']}
  8. Proof of \ref{['thm1']}
  9. Proof of \ref{['thm2']} using \ref{['thm1']}
  10. Proof of \ref{['thm3']}
  11. Concluding Remarks

Key Result

Theorem 1

(Turán Turan) Let $G$ be a simple graph on $n$ vertices with clique number at most $r$. Then with equality if and only if $G$ is a regular Turán graph on $n$ vertices with $r$ classes.

Figures (2)

  • Figure 1: $E(f_1) = \{e_1,e_2,e_3\}$, thus $d(f_1) = 3$. But $E(f_3) = \{e_1,e_2,e_4,e_5,e_6,e_7,e_4,e_3\}$, note that $e_4$ comes twice, and therefore $d(f_3) = 8$.
  • Figure 2: Diamond in the extremal graph $G$

Theorems & Definitions (37)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Theorem 4
  • Corollary 2
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 27 more