Localization: A Framework to Generalize Extremal Graph Problems

Rajat Adak; L. Sunil Chandran

arXiv:2508.20946·math.CO·March 10, 2026

Localization: A Framework to Generalize Extremal Graph Problems

Rajat Adak, L. Sunil Chandran

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TL;DR

This paper introduces a localization framework to improve bounds in extremal graph theory and characterizes the structure of extremal graphs for various problems.

Contribution

It develops a new localization approach that refines classical extremal bounds and provides structural insights into extremal graphs.

Findings

01

Improved bounds for the number of cliques in graphs with degree constraints.

02

Structural characterizations of extremal graphs achieving these bounds.

03

Generalization of classical extremal results using localization.

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,…

Equations114

m \leq \frac{g}{g - 2} (n - 2)

m \leq \frac{g}{g - 2} (n - 2)

e x (n, K_{t}, K_{1, d + 1}) \leq \frac{n}{d + 1} (t d + 1)

e x (n, K_{t}, K_{1, d + 1}) \leq \frac{n}{d + 1} (t d + 1)

m e x (m, K_{t}, K_{1, d + 1}) \leq \frac{m}{( 2 d + 1 )} (t d + 1)

m e x (m, K_{t}, P_{r + 1}) \leq \frac{m}{( 2 r )} (t r)

m e x (m, K_{t}, P_{r + 1}) \leq \frac{m}{( 2 r )} (t r)

∣ E (G) ∣ \leq \frac{n ^{2} ( r - 1 )}{2 r},

∣ E (G) ∣ \leq \frac{n ^{2} ( r - 1 )}{2 r},

∣ E (G) ∣ \leq \frac{g}{g - 2} (n - 2)

∣ E (G) ∣ \leq \frac{g}{g - 2} (n - 2)

N (G, K_{t}) \leq e x (n, K_{t}, K_{1, d + 1}) \leq \frac{n}{d + 1} (t d + 1) = \frac{n}{t} (t - 1 d)

N (G, K_{t}) \leq e x (n, K_{t}, K_{1, d + 1}) \leq \frac{n}{d + 1} (t d + 1) = \frac{n}{t} (t - 1 d)

N (G, K_{\geq 1}) \leq \frac{n}{d + 1} (2^{d + 1} - 1) .

N (G, K_{\geq 1}) \leq \frac{n}{d + 1} (2^{d + 1} - 1) .

N (G, K_{t}) \leq m e x (m, K_{t}, K_{1, d + 1}) \leq \frac{m}{( 2 d + 1 )} (t d + 1) = \frac{m}{( 2 t )} (t - 2 d - 1)

N (G, K_{t}) \leq m e x (m, K_{t}, K_{1, d + 1}) \leq \frac{m}{( 2 d + 1 )} (t d + 1) = \frac{m}{( 2 t )} (t - 2 d - 1)

N (G, K_{t \geq 2}) \leq \frac{m}{( 2 d + 1 )} (2^{d + 1} - d - 2) .

N (G, K_{t \geq 2}) \leq \frac{m}{( 2 d + 1 )} (2^{d + 1} - d - 2) .

N (G, K_{t}) \leq m e x (m, K_{t}, P_{r + 1}) \leq \frac{m}{( 2 r )} (t r)

N (G, K_{t}) \leq m e x (m, K_{t}, P_{r + 1}) \leq \frac{m}{( 2 r )} (t r)

α (G) \geq \frac{n}{Δ + 1},

α (G) \geq \frac{n}{Δ + 1},

α (G) \geq v \in V (G) \sum \frac{1}{d ( v ) + 1},

α (G) \geq v \in V (G) \sum \frac{1}{d ( v ) + 1},

e \in E (G) \sum \frac{w ( e )}{w ( e ) - 1} \leq \frac{n ^{2}}{2},

e \in E (G) \sum \frac{w ( e )}{w ( e ) - 1} \leq \frac{n ^{2}}{2},

∣ E (G) ∣ \leq \frac{n}{2} v \in V (G) \sum \frac{c ( v ) - 1}{c ( v )}

∣ E (G) ∣ \leq \frac{n}{2} v \in V (G) \sum \frac{c ( v ) - 1}{c ( v )}

e \in E (G) \sum \frac{g ( e ) - 2}{g ( e )} \leq n - 2

e \in E (G) \sum \frac{g ( e ) - 2}{g ( e )} \leq n - 2

N (G, K_{t}) \leq v \in V (G) \sum \frac{( t d ( v ) + 1 )}{d ( v ) + 1} = \frac{1}{t} v \in V (G) \sum (t - 1 d ( v ))

N (G, K_{t}) \leq v \in V (G) \sum \frac{( t d ( v ) + 1 )}{d ( v ) + 1} = \frac{1}{t} v \in V (G) \sum (t - 1 d ( v ))

N (G, K_{t \geq 1}) \leq v \in V (G) \sum t = 1 \sum d (v) + 1 \frac{( t d ( v ) + 1 )}{d ( v ) + 1} = v \in V (G) \sum \frac{2 ^{d (v) + 1} - 1}{d ( v ) + 1} .

N (G, K_{t \geq 1}) \leq v \in V (G) \sum t = 1 \sum d (v) + 1 \frac{( t d ( v ) + 1 )}{d ( v ) + 1} = v \in V (G) \sum \frac{2 ^{d (v) + 1} - 1}{d ( v ) + 1} .

N (G, K_{t}) \leq e \in E (G) \sum \frac{( t w ( e ) + 2 )}{( 2 w ( e ) + 2 )} = \frac{1}{( 2 t )} e \in E (G) \sum (t - 2 w ( e ))

N (G, K_{t}) \leq e \in E (G) \sum \frac{( t w ( e ) + 2 )}{( 2 w ( e ) + 2 )} = \frac{1}{( 2 t )} e \in E (G) \sum (t - 2 w ( e ))

N (G, K_{t \geq 2}) \leq e \in E (G) \sum t = 2 \sum w (e) + 2 \frac{( t w ( e ) + 2 )}{( 2 w ( e ) + 2 )} = e \in E (G) \sum \frac{2 ^{w (e) + 2} - w ( e ) - 3}{( 2 w ( e ) + 2 )} .

N (G, K_{t \geq 2}) \leq e \in E (G) \sum t = 2 \sum w (e) + 2 \frac{( t w ( e ) + 2 )}{( 2 w ( e ) + 2 )} = e \in E (G) \sum \frac{2 ^{w (e) + 2} - w ( e ) - 3}{( 2 w ( e ) + 2 )} .

N (G, K_{t}) \leq e \in E (G) \sum \frac{( t p ( e ) + 1 )}{( 2 p ( e ) + 1 )} = \frac{1}{( 2 t )} e \in E (G) \sum (t - 2 p ( e ) - 1)

N (G, K_{t}) \leq e \in E (G) \sum \frac{( t p ( e ) + 1 )}{( 2 p ( e ) + 1 )} = \frac{1}{( 2 t )} e \in E (G) \sum (t - 2 p ( e ) - 1)

n - 2 \geq e \in E (G) \sum \frac{g ( e ) - 2}{g ( e )} \geq e \in E (G) \sum \frac{g - 2}{g} = \frac{g - 2}{g} m

n - 2 \geq e \in E (G) \sum \frac{g ( e ) - 2}{g ( e )} \geq e \in E (G) \sum \frac{g - 2}{g} = \frac{g - 2}{g} m

⟹ m \leq \frac{g}{g - 2} (n - 2)

N (G, K_{t}) \leq \frac{1}{t} v \in V (G) \sum (t - 1 d ( v )) \leq \frac{1}{t} v \in V (G) \sum (t - 1 d) = \frac{n}{t} (t - 1 d)

N (G, K_{t}) \leq \frac{1}{t} v \in V (G) \sum (t - 1 d ( v )) \leq \frac{1}{t} v \in V (G) \sum (t - 1 d) = \frac{n}{t} (t - 1 d)

N (G, K_{t}) \leq \frac{1}{( 2 t )} e \in E (G) \sum (t - 2 w ( e )) \leq \frac{m}{( 2 t )} (t - 2 d - 1)

N (G, K_{t}) \leq \frac{1}{( 2 t )} e \in E (G) \sum (t - 2 w ( e )) \leq \frac{m}{( 2 t )} (t - 2 d - 1)

N (G, K_{t}) \leq \frac{1}{( 2 t )} e \in E (G) \sum (t - 2 p ( e ) - 1) \leq e \in E (G) \sum \frac{( t r )}{( 2 r )} = \frac{m}{( 2 r )} (t r)

N (G, K_{t}) \leq \frac{1}{( 2 t )} e \in E (G) \sum (t - 2 p ( e ) - 1) \leq e \in E (G) \sum \frac{( t r )}{( 2 r )} = \frac{m}{( 2 r )} (t r)

∣ F (G) ∣ = 2 - ∣ V (G) ∣ + ∣ E (G) ∣

∣ F (G) ∣ = 2 - ∣ V (G) ∣ + ∣ E (G) ∣

2 e \in E (G) \sum \frac{1}{g ( e )} \geq i = 1 \sum ∣ F (G) ∣ e \in E (f_{i}) \sum \frac{1}{d ( f _{i} )} = i = 1 \sum ∣ F (G) ∣ d (f_{i}) \frac{1}{d ( f _{i} )} = ∣ F (G) ∣

2 e \in E (G) \sum \frac{1}{g ( e )} \geq i = 1 \sum ∣ F (G) ∣ e \in E (f_{i}) \sum \frac{1}{d ( f _{i} )} = i = 1 \sum ∣ F (G) ∣ d (f_{i}) \frac{1}{d ( f _{i} )} = ∣ F (G) ∣

2 e \in E (G) \sum \frac{1}{g ( e )} \geq 2 - ∣ V (G) ∣ + ∣ E (G) ∣

2 e \in E (G) \sum \frac{1}{g ( e )} \geq 2 - ∣ V (G) ∣ + ∣ E (G) ∣

⟹

N (G, K_{t}) = N (G^{'}, K_{t}) + N (G, K_{t}, x)

N (G, K_{t}) = N (G^{'}, K_{t}) + N (G, K_{t}, x)

N (G, K_{t}, x) = N (G [N (x)], K_{t - 1}) \leq (t - 1 d ( x ))

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[1]\fnmRajat \surAdak\orcidhttps://orcid.org/0009-0001-2723-5550 1]\orgdivDepartment of Computer Science and Automation, \orgnameIndian Institute of Science, \orgaddress\cityBangalore, \countryIndia

Localization: A Framework to Generalize Extremal Graph Problems

[email protected]

\fnmL. Sunil \surChandran

[email protected]

[

Abstract

Extremal graph theory studies the maximum or minimum number of subgraphs isomorphic to a prescribed graph under given constraints. 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

[TABLE]

Wood [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:

[TABLE]

More recently, Chakraborty and Chen [CHAKRABORTI2024103955] established a similar upper bound for graphs with bounded path length:

[TABLE]

In this paper, we employ the localization framework to improve these bounds and provide structural characterizations of the extremal graphs attaining them.

keywords:

Extremal graph theory, Localization, Planar graphs, Cliques, Turán number

1 Introduction

Extremal graph theory, initiated by Turán, studies the maximum or minimum number of copies of a certain subgraph in a graph that does not contain any member of a prescribed family as a subgraph. Some of the earliest foundational results in extremal graph theory are the following classical theorems.

Theorem 1.

(Turán [Turan])* Let $G$ be a simple graph on $n$ vertices with clique number at most $r$ . Then*

[TABLE]

with equality if and only if $G$ is a regular Turán graph on $n$ vertices with $r$ classes.

Definition 1.

A planar graph $G$ is called a $k$ -angulation if it admits a planar embedding in which every face (including the outer face) is a cycle of length $k$ .

Theorem 2.

Let $G$ be a simple planar connected graph on $n$ vertices with finite girth $g$ , where girth is the length of the smallest cycle in the graph. Then

[TABLE]

with equality if and only if $G$ is a $g$ -angulation.

Let $\mathcal{F}$ be a family of graphs. A graph is considered $\mathcal{F}$ -free if it contains no subgraph that is isomorphic to any graph in the family $\mathcal{F}$ . Turán number of an $\mathcal{F}$ -free graph on $n$ vertices is denoted by $ex(n,\mathcal{F})$ , which counts the maximum number of edges in such a graph. The generalized Turán numbers $ex(n,H,\mathcal{F})$ and $mex(m,H,\mathcal{F})$ extend this framework by counting the maximum number of copies of a fixed graph $H$ that can appear in $\mathcal{F}$ -free graphs on $n$ vertices or with $m$ edges, respectively. For the special case $\mathcal{F}=\{F\}$ , we simplify the notation to $ex(n,H,F)$ (or $mex(m,H,F)$ ). These parameters have been widely studied when $\mathcal{F}$ is a natural family of graphs, such as paths, cycles, and stars and $H$ is a clique.

Notation 1.

Let $N(G,K_{t})$ denote the number of subgraphs of $G$ isomorphic to $K_{t}$ .

Theorem 3.

(Wood [wood])* Let $t\geq 1$ and $G$ be a graph on $n$ vertices with maximum degree, $\Delta(G)=d$ . Then;*

[TABLE]

If $t=1$ , we always get equality; otherwise, equality holds if and only if $G$ is a disjoint union of copies of $K_{d+1}$ .

Corollary 1.

(Wood [wood])* Let $t\geq 1$ and $G$ be a graph on $n$ vertices with maximum degree $\Delta(G)=d$ . Then*

[TABLE]

Theorem 4.

(Wood [wood])* Let $t\geq 2$ and $G$ be a graph with $m$ edges and maximum degree, $\Delta(G)=d$ . Then;*

[TABLE]

If $t=2$ , we always get equality; otherwise, equality holds if and only if $G$ is a disjoint union of copies of $K_{d+1}$ with any number of isolated vertices.

Corollary 2.

(Wood [wood])* Let $t\geq 2$ and $G$ be a graph with $m$ edges and maximum degree $\Delta(G)=d$ . Then*

[TABLE]

Theorem 5.

(Chakraborti-Chen [CHAKRABORTI2024103955])* Let $t\geq 2$ and $G$ be a $P_{r+1}$ -free graph on $m$ edges, then*

[TABLE]

If $t=2$ , equality always holds; otherwise, equality holds if and only if $G$ is a disjoint union of copies of $K_{r}$ with any number of isolated vertices.

1.1 Localization

A classical lower bound on the independence number $\alpha(G)$ of a graph $G$ is

[TABLE]

where $\Delta$ denotes the maximum degree of $G$ . While this inequality depends only on a global parameter, namely the maximum degree, stronger bounds can often be obtained by incorporating local structural information. In this direction, Caro [caro1979new] and Wei [wei1981lower] independently established a celebrated refinement by replacing $\Delta$ with the degree of each individual vertex. Their result asserts that

[TABLE]

where $d(v)$ denotes the degree of vertex $v\in V(G)$ . This formulation highlights the power of localized parameters in strengthening classical extremal results, and naturally, this led to the question of whether other graph parameters, when localized, could provide comparable extensions of classical extremal theorems.

Bradač [bradac] and Malec–Tompkins [DBLP:journals/ejc/MalecT23] recently introduced the notion of localization as a tool to strengthen and generalize classical extremal bounds. Observe that the constraint in Theorem˜1 relies on a global graph parameter—specifically, the maximum clique size. In contrast, the localization framework assigns local weights to graph elements (such as vertices or edges), thereby transforming global restrictions into locally defined conditions.

Bradač [bradac] and Malec–Tompkins [DBLP:journals/ejc/MalecT23] defined a weight function $w:E(G)\rightarrow\mathbb{Z}$ , where the value of $w(e)$ is determined by the particular extremal bound under consideration. In the case of Theorem˜1, $w(e)$ is defined as the order of the largest clique in $G$ containing the edge $e$ . They proved the following result:

Theorem 6.

(Bradač [bradac], Malec–Tompkins [DBLP:journals/ejc/MalecT23])* For a simple graph $G$ with $n$ vertices,*

[TABLE]

with equality if and only if $G$ is a regular Turán graph.

It is easy to verify that Theorem˜6 implies the bound of Theorem˜1, when we assume $G$ to be $K_{r+1}$ -free. These localized weights allow for the improvement of the classical bound in Theorem˜1 by replacing the global parameter, maximum clique size, with these edge weights.

In [zhao2025localized], the localization framework was further applied to generalize the Erdős–Gallai theorem for cycles [gallai1959maximal] and the corresponding path theorem was localized in [DBLP:journals/ejc/MalecT23]. Kirsch and Nir [Kirsch_2024] established a localized version of Zykov’s generalization [Zykov1949-qr] of Turán’s theorem, which was subsequently further improved in [aragao2024localised], the authors provided a localization for the Graph Maclaurin Inequalities [khadzhiivanov1977inequalities]. Moreover, the localization framework has also been extended to hypergraphs [zhao2025localizedhyper] and to spectral extremal bounds [liu2026local, kannan2025localizationspectralturantypetheorems]. This further illustrates the versatility and potential of the localization framework to extend across different domains within extremal graph theory.

While edge-based localization often produces distribution-type inequalities reminiscent of the LYM or Kraft inequalities, a more direct alternative was proposed in [adak2025vertex], where localization is applied to vertices rather than edges.

Adak and Chandran [adak2025vertex] introduced the concept of vertex-based localization. They provided a vertex-based localization of the Erdős–Gallai theorems for paths and cycles. In subsequent work [adak2025turan], the same authors established a vertex-based localization of a popular form of Turán’s theorem. For a simple graph $G$ they proved that,

[TABLE]

where the weight function $c:V(G)\rightarrow\mathbb{Z}$ is defined such that $c(v)$ is the order of the largest clique containing the vertex $v$ .

More recently, in [adak2025vertexbasedlocalizationgeneralizedturan], they extended this localization framework to improve the bounds of generalized Erdős–Gallai theorems due to Luo [luo2018maximum] and in [adak2025generalizedzykovstheorem] they provided a vertex-based localization of Zykov’s Theorem.

In this paper, we use the localization framework to generalize Theorem˜2,3, 4 and 5, and characterize the extremal graphs for these generalized bounds.

2 Main Results

We provide a vertex-based localization of Theorem˜3, and generalize Theorems˜2, 4 and 5, by assigning suitable weights to the edges of the graph and applying the localization framework.

Notation 2.

Define $g:E(G)\rightarrow\mathbb{Z}$ , such that $g(e)$ is the length of the smallest cycle edge $e$ is part of, and if $e$ does not participate in any cycle, set $g(e)=2$ .

Theorem 7.

Let $G$ be a planar connected graph on $n$ vertices. Then;

[TABLE]

equality holds if and only if $G$ is either $K_{2}$ or is a $k$ -angulation for some $k$ .

Theorem 8.

Let $t\geq 1$ and $G$ be a simple graph then,

[TABLE]

Let $X=\{v\in V(G)\mid d(v)\geq t-1\}$ . If $t=1$ , equality always holds; otherwise, equality holds if and only if all the components of $G[X]$ are cliques.

Corollary 3.

Let $t\geq 1$ and $G$ be a simple graph. Then

[TABLE]

The above result can be seen as a localized version of Corollary˜1.

Definition 2.

A graph $G$ is called a diamond-graph if it is isomorphic to $K_{4}$ minus one edge. A graph is diamond-free if it has no diamond as an induced subgraph.

Notation 3.

Define $w:E(G)\rightarrow\mathbb{Z}$ , such that $w(e)=$ Number of common neighbors of the endpoints of $e$ . In other words, $w(e)$ is the number of triangles that edge $e$ is part of in $G$ . This can be seen as the degree of edge $e$ .

Theorem 9.

Let $t\geq 2$ and $G$ be a simple graph then,

[TABLE]

Let $Y=\{e\in E(G)\mid w(e)<t-2\}$ . If $t=2$ , equality always holds; otherwise, equality holds if and only if $G\setminus Y$ is a diamond-free graph.

Corollary 4.

Let $t\geq 2$ and $G$ be a simple graph. Then

[TABLE]

Corollary˜4 can be seen as a localized version of Corollary˜2.

Notation 4.

Define $p:E(G)\rightarrow\mathbb{Z}$ , such that $p(e)$ is the length of the longest path in $G$ containing the edge $e$ .

Theorem 10.

Let $t\geq 2$ and $G$ be a simple graph, then

[TABLE]

Let $Z=\{e\in E(G)\mid p(e)<t-1\}$ . If $t=2$ , equality always holds; otherwise, equality holds if and only if all the components of $G\setminus Z$ are cliques.

2.1 Recovering the classical bounds

From Theorem˜7 to Theorem˜2 (For $2$ -connected graphs)

Assume that $G$ is $2$ -connected and has a finite girth $g$ . Thus there are no cut-edges in $G$ . Therefore, $g\leq g(e)$ for all $e\in E(G)$ . Note that $g>2$ . Therefore, from the bound of Theorem˜7 we get;

[TABLE]

For equality, we must have equality in the bound of Theorem˜7. Thus $G$ must be a $K_{2}$ or a $k$ -angulation. Since $G$ has finite girth $G\not\cong K_{2}$ and since the girth of $G$ is $g$ , we get that $k=g$ . Thus $G$ is $g$ -angulation.

From Theorem˜8 to Theorem˜3

Since $\Delta(G)=d$ , therefore $d\geq d(v)$ for all $v\in V(G)$ . Thus $\frac{1}{t}{d(v)\choose t-1}\leq\frac{1}{t}{d\choose t-1}$ for all $v\in V(G)$ . Therefore, from the bound of Theorem˜8 we get;

[TABLE]

For equality, both the above inequalities must be equalities. Thus $G$ must be $d$ -regular graph. From Theorem˜8, the equality holds only if all the components of $G[X]$ are cliques. Therefore, $G$ must be a disjoint union of copies of $K_{d+1}$ .

From Theorem˜9 to Theorem˜4

Since $\Delta(G)=d$ , for all $e=\{u,v\}\in E(G)$ , $w(e)\leq\text{min}\{d(v),d(u)\}-1\leq\Delta(G)-1=d-1$ . Therefore, for all $e\in E(G)$ we have, $\frac{1}{{t\choose 2}}{w(e)\choose t-2}\leq\frac{1}{{t\choose 2}}{d-1\choose t-2}$ . Thus, from the bound of Theorem˜9 we get;

[TABLE]

For equality, we must have $w(e)=d-1$ for all $e\in E(G)$ . Since $w(e)\leq\text{min}\{d(v),d(u)\}-1\leq d-1$ , for all $x\in V(G)$ such that there exists an edge adjacent to $x$ , that is, $d(x)>0$ , we get that, $d(x)=d$ . Also from theorem˜9, $G\setminus Y$ must be a diamond-free graph. Thus $G\setminus Y$ must be a disjoint union of copies of $K_{d+1}$ and since $w(e)=d-1$ for all $e\in E(G)$ and $d(x)=d$ for all $x\in V(G\setminus Y)$ . Note that $Y$ must consist of only isolated vertices. Thus $G$ is a disjoint union of copies of $K_{d+1}$ and isolated vertices.

From Theorem˜10 to Theorem˜5

Since $G$ is $P_{r+1}$ -free, we get that $p(e)+1\leq r$ for all $e\in E(G)$ . Since $p(e)+1\geq 2$ we get that, $\frac{1}{{t\choose 2}}{p(e)-1\choose t-2}\leq\frac{1}{{t\choose 2}}{r-2\choose t-2}=\frac{{r\choose t}}{{r\choose 2}}$ . Therefore, from the bound of Theorem˜5 we get;

[TABLE]

For equality, both the above inequalities must be equalities. From Theorem˜10, the equality holds only if all the components of $G\setminus Z$ are cliques. Also, $p(e)+1=r$ for all $e\in E(G)$ . Thus, the cliques are either of order $r$ or $1$ .

3 Proof of Results

3.1 Proof of Theorem˜7

Proof.

Since $G$ is planar and connected, from Euler’s formula, we get that;

[TABLE]

Let $F(G)=\{f_{1},f_{2},\dots,f_{|F(G)|}\}$ . If an edge $e\in E(G)$ is incident to a face $f_{i}$ for some $i\in[\,|F(G)|\,]$ , then $g(e)\leq d(f_{i})$ , where $d(f_{i})$ denotes the degree of the face $f_{i}$ , that is the number of edges in the boundary walk of $f_{i}$ (thus cut-edges are counted twice).

Observe that every edge of $G$ which is not a cut edge is incident to exactly two faces, contributing once to the degree of each of them. In contrast, a cut-edge is incident to a single face but contributes twice to its degree.

Let $E(f_{i})$ be the multiset of edges in the boundary walk of $f_{i}$ , where the cut-edges in the boundary of $f_{i}$ , appear twice. Refer to fig.˜1 for an example. Note that $|E(f_{i})|=d(f_{i})$ . Thus, we get;

[TABLE]

Thus from eqs.˜1 and 2 we get that;

[TABLE]

For the extremal analysis, we first consider the case when $G$ is either $K_{2}$ or a $k$ -angulation. If $G\cong K_{2}$ , then $E(G)=\{e\}$ and $g(e)=2$ . In this case, both sides of the bound evaluate to [math]. If $G$ is a $k$ -angulation, it follows that $g(e)=k$ for all $e\in E(G)$ and $d(f_{i})=k$ for all $f_{i}\in F(G)$ . Hence, equality holds in eq.˜2, and consequently, in the bound of Theorem˜7.

For the converse, assume equality holds in the bound of Theorem˜7 with $G\not\cong K_{2}$ . This implies equality in eq.˜2, and therefore $d(f_{i})=g(e)$ for every $f_{i}\in F(G)$ and every $e\in E(f_{i})$ . Since $g(e)=2$ if and only if $e$ is a cut-edge, while $d(f_{i})>2$ for all $f_{i}\in F(G)$ (as $G\not\cong K_{2}$ ), it follows that $G$ contains no cut-edges and therefore, each face in $G$ is bounded by a cycle.

Two faces are adjacent if they share at least one edge. In the dual graph, this corresponds to the adjacency of the associated vertices. Similarly, two faces are connected if the corresponding vertices in the dual graph are connected.

If $f_{i}$ and $f_{j}$ are adjacent and share an edge $e$ , then we have $d(f_{i})=d(f_{j})=g(e)$ . Since the dual graph of any planar graph is connected, every face is connected to every other face. By propagating this equality of $d(\cdot)$ along paths in the dual, it follows that all faces share the same degree. Hence $d(f)$ is constant for all $f\in F(G)$ .

Since every face is bounded by a cycle, the existence of a face $f$ whose boundary cycle has length strictly less than $d(f)$ would imply the presence of an edge $e$ on the boundary of $f$ with $g(e)<d(f)$ . This is a contradiction. Hence, each face must be bounded by a simple cycle of length exactly $d(f)$ . Consequently, $G$ is a $k$ -angulation with $k=d(f)$ for any $f\in F(G)$ . ∎

3.2 Proof of Theorem˜8

Notation 5.

If $v\in V(G)$ , then $N(G,K_{t},v)$ denotes the number of copies of $K_{t}$ in $G$ containing the vertex $v$ .

Recall that the set $X$ consists of all vertices in $G$ whose degrees are at least $t-1$ .

Lemma 1.

Theorem˜8* holds for $G$ is and only if it holds for $G\setminus X$ .*

Proof.

This follows from the fact that vertices in the complement set $X^{c}=V(G)\setminus X$ do not contribute to either side of the inequality in Theorem˜8. Note that $N(G,K_{t})=N(G[X],K_{t})$ , since any clique of order at least $t$ in $G$ , that is not entirely in $G[X]$ has to contain a vertex $v\in X^{c}$ ; but then $d(v)\geq t-1$ , contradiction. On the other hand, since for $v\in X^{c}$ , we have $d(v)<t-1$ , we get ${d(v)\choose t-1}=0$ . ∎

Therefore, without loss of generality, we may assume that $G$ contains only vertices of degree at least $t-1$ . Now we are ready to prove Theorem˜8.

Proof.

For $t=1$ , the bound and the equality case can both be verified trivially; thus, we assume $t\geq 2$ . Since we assumed $d(v)\geq t-1$ for all $v\in V(G)$ , we get that $G$ does not contain any isolated vertex; therefore, $\delta(G)>0$ .

Proof by induction on the number of vertices.

Base Case: For $n=2$ , let $V(G)=\{u,v\}$ and $E(G)=\{\{u,v\}\}$ (since $\delta(G)>0)$ . For $t>2$ , the bound is trivially true since both sides are zero. For $t=2$ , $N(G,K_{t})=1$ and the right-hand side also sums up to $1$ .

Induction Hypothesis: Suppose the claim in Theorem˜8 is true for all graphs with less than $|V(G)|$ vertices.

Induction Step: Let $x\in V(G)$ such that $0<d(x)=\delta(G)$ . Now define $G^{\prime}=G\setminus\{x\}$ . Recall that $N(G,K_{t},x)$ denotes the number of copies of $K_{t}$ in $G$ containing the vertex $x\in V(G)$ . Clearly;

[TABLE]

Note that for every $K_{t}$ containing $x$ , there exists a unique copy of $K_{t-1}$ in $G[N(x)]$ . Therefore;

[TABLE]

Let $d_{G^{\prime}}(v)$ be the degree of vertex $v$ in $G^{\prime}$ . Now note that $d_{G^{\prime}}(v)=d(v)$ for all $v\in V(G^{\prime})\setminus N(x)$ , where $N(x)$ denotes the set of vertices adjacent to $x$ in $G$ . Clearly, $d_{G^{\prime}}(v)=d(v)-1$ for all $v\in N(x)$ . From the induction hypothesis, we know that;

[TABLE]

Thus from eqs.˜3, 4 and 5 we get that;

[TABLE]

Note that;

[TABLE]

Thus from eqs.˜6 and 7 we get;

[TABLE]

Claim 1.

For $d(v)\geq d(x)>0$ and $t\geq 2$ we have;

[TABLE]

Proof.

Note that $\delta(G)=d(x)\leq d(v)$ for all $v\in V(G)$ . Therefore, we get $\frac{1}{t-1}{d(v)-1\choose t-2}\geq\frac{1}{t-1}{d(x)-1\choose t-2}\implies\frac{1}{d(v)}{d(v)\choose t-1}\geq\frac{1}{d(x)}{d(x)\choose t-1}$ . Thus we get;

[TABLE]

∎

From sections˜3.2 and 1 we get that;

[TABLE]

Thus, we get the required bound. Now we will characterize the extremal graphs for Theorem˜8.

First, suppose that all the components of $G$ are cliques. Let $C$ be one such component with order $k$ . Clearly $d(v)=k-1$ for all $v\in V(C)$ . And $C$ contains ${k\choose t}$ copies of $K_{t}$ . The contribution by the vertices of $C$ to the left side of the bound in Theorem˜8 is,

[TABLE]

Thus, we get equality in the bound.

Now, assume that we have equality in the bounds of Theorem˜8. Thus, we must have equality in eqs.˜4, 5 and 9. From equality in eq.˜4, we get that $G[N[x]]$ is a clique, where $N[x]$ denotes the closed neighborhood of $x$ . Using the induction hypothesis and the equality in eq.˜5, we get that all the components of $G^{\prime}$ are cliques. Assume that $C$ is a connected component of $G$ , then it is enough to show that $C$ is a clique.

If $C$ does not contain $x$ , clearly $C$ does not contain any vertex from $N(x)$ . Thus, $C$ is a component in $G^{\prime}$ as well, and therefore, by induction hypothesis, $C$ is a clique. Now suppose $C$ contains the vertex $x$ , therefore $C$ contains all the vertices of $N(x)$ . Since $N[x]$ induces a clique in $G$ , $C\setminus\{x\}$ must be connected and therefore, $C\setminus\{x\}$ must be a component of $G^{\prime}$ , and thus is a clique. Suppose there exists $y\in V(C)\setminus N[x]$ . Clearly $y$ is adjacent to $v\in N(x)$ but not adjacent to $x$ . Therefore, $d(v)>d(x)$ . But from equality in the bound of eq.˜9 we get that, $d(x)=d(v)$ for all $v\in N(x)$ . We get a contradiction. Therefore, $V(C)=N[x]$ , which induces a clique. ∎

3.3 Proof of Theorem˜9

Recall that $Y=\{e\in E(G)\mid w(e)+2<t\}$

Lemma 2.

Theorem˜9* holds for $G$ if and only if it holds for $G\setminus Y$ *

Proof.

This follows from the fact that edges in the set $Y$ do not contribute to either side of the inequality in Theorem˜9. Note that for any edge $e\in Y$ , the endpoints of $e$ have fewer than $t-2$ common neighbors in $G$ . Hence, no copy of $K_{t}$ in $G$ can contain $e$ . On the other hand, since $w(e)<t-2$ , we have ${w(e)\choose t-2}=0$ . ∎

Therefore, without loss of generality, we may assume that $G$ contains only edges with weights at least $t-2$ . Now coming to the proof of Theorem˜9.

Proof.

If $t=2$ , clearly the bound holds with equality; thus, assume $t\geq 3$ . Let $e\in E(G)$ , if $e=\{u,v\}$ where $u,v\in V(G)$ . If $A$ is a $K_{t}$ containing the edge $e$ , then $u,v\in V(A)$ and $V(A)\setminus\{u,v\}\subseteq N(u)\cap N(v)$ . Note that $|N(u)\cap N(v)|=w(e)$ . Thus, the number of copies of $K_{t}$ containing the edge $e$ is at most ${w(e)\choose t-2}$ . If we add up the contribution of each edge, that is the number of copies of $K_{t}$ , containing an edge, then each copy is counted ${t\choose 2}$ times, since it is counted once for each edge participating in that $K_{t}$ copy. Thus, we get;

[TABLE]

Thus, we get the required bound. Now we will characterize the extremal graphs for Theorem˜9.

First, assume we have equality in Theorem˜9. Therefore, we must have the number of copies of $K_{t}$ containing the edge $e$ to be exactly ${w(e)\choose t-2}$ for all $e\in E(G)$ . Thus, the vertices in $N(u)\cap N(v)$ must induce a clique where $u,v\in V(G)$ are the endpoints of $e$ . Now suppose $G$ contains an induced diamond as shown in fig.˜2. Note that $\{a,b\}\subseteq N(c)\cap N(d)$ , but $a$ and $b$ are not adjacent. Thus, we get a contradiction; therefore, $G$ is diamond-free.

Now suppose $G$ is diamond-free. Suppose there exists an edge $e=\{x,y\}\in E(G)$ such that $N(x)\cap N(y)$ does not induce a clique in $G$ . Since $N(x)\cap N(y)$ is non-empty, there exists $u,v\in N(x)\cap N(y)$ such that $u$ and $v$ are not adjacent. Therefore, we get a diamond induced by $u,v,x$ , and $y$ . Thus, the common neighbors of the endpoints of any edge in $G$ must induce a clique. Therefore, the number of copies of $K_{t}$ containing some $e\in E(G)={w(e)\choose t-2}$ . Therefore, we get $G$ must be extremal for Theorem˜9. ∎

3.4 Proof of Theorem˜8 using Theorem˜9

While an independent proof of Theorem˜8 was presented in section˜3.2, we now offer an alternative derivation using Theorem˜9. This does not diminish the value of the original argument, as the degree function $d(v)$ for $v\in V(G)$ is a more natural and widely adopted parameter compared to the edge-degree parameter, $w(e)$ for $e\in E(G)$ .

Proof.

For $t=1$ , it is easy to verify that the bound holds with equality; thus, we will assume $t\geq 2$ . Let $e\in E(G)$ such that $\{u,v\}=e$ where $u,v\in V(G)$ . Clearly $w(e)\leq d(u)-1$ and $w(e)\leq d(v)-1$ . Note that the number of edges in $E(G)$ which have a vertex $x\in V(G)$ as one endpoint is $d(x)$ . Thus from Theorem˜9 we get;

[TABLE]

Thus, we obtain the desired inequality. Observe that equality holds if and only if both inequalities in eqs.˜11 and 12 are tight. Specifically, equality in eq.˜11 holds if and only if $G$ is diamond-free, while equality in eq.˜12 holds if and only if for every edge $e=\{u,v\}\in E(G)$ , we have $w(e)=d(u)-1=d(v)-1$ .

Note that if $G$ is a disjoint union of cliques, it is straightforward to verify that the above inequality holds with equality.

For the converse, assume equality holds in eqs.˜11 and 12. Let $C$ be a connected component of $G$ ; it suffices to show that $C$ is a clique. Let $K$ be a maximum clique in $C$ . Clearly, $|V(K)|>1$ . Suppose $V(C)\setminus V(K)\neq\emptyset$ . Then there exists some $v\in V(K)$ adjacent to a vertex $x\in V(C)\setminus V(K)$ . Choose $u\in V(K)$ with $u\neq v$ . Since $w(\{u,v\})=d(v)-1=d(u)-1$ , we have $N[u]=N[v]$ . In particular, $x$ is adjacent to $u$ .

Thus, $x,u,v$ form a triangle, and hence $|V(K)|\geq 3$ . Consequently, there exists some $z\in V(K)$ not adjacent to $x$ ; otherwise, $K\cup\{x\}$ would induce a larger clique in $C$ , contradicting the maximality of $K$ . But then $x,u,v,z$ induce a diamond in $C$ , and hence in $G$ , contradicting the equality condition in eq.˜11. Therefore, $V(C)\setminus V(K)=\emptyset$ , and $C$ must be a clique. ∎

3.5 Proof of Theorem˜10

Recall that $Z=\{e\in E(G)\mid p(e)+1<t\}$ .

Lemma 3.

Theorem˜10* holds for $G$ if and only if it holds for $G\setminus Z$ .*

Proof.

This follows from the fact that edges in the set $Z$ do not contribute to either side of the inequality in Theorem˜10. Note that for any edge $e\in Z$ , the longest path containing $e$ has length less than $t-1$ . Hence, no copy of $K_{t}$ in $G$ can contain $e$ . On the other hand, since $p(e)<t-1$ , we have $p(e)-1<t-2$ , and therefore ${p(e)-1\choose t-2}=0$ . ∎

Therefore, without loss of generality, we may assume $p(e)\geq t-1$ for all $e\in E(G)$ .

Proof.

If $t=2$ , clearly the bound holds with equality; thus, assume $t\geq 3$ .

Proof by induction on the number of vertices of the graph.

Base Case: For $n=1$ , then there are no edges and the claim is vacuously true.

For $n=2$ and $|E(G)|=1$ (For $|E(G)|=0$ , the claim is again vacuously true), say $E(G)=\{e\}$ . Note that $p(e)=1$ . Since $t>2$ , clearly both sides of the bound are zero. For $n=3$ , let $V(G)=\{u,v,w\}$ and $\{\{u,v\},\{v,w\}\}\subseteq E(G)$ (since $p(e)\geq t-1\geq 2)$ . If $t>3$ , the bound is trivially true as both sides are zero. If $t=3$ , $N(G,K_{t})\leq 1$ , with equality if and only if $G\cong K_{3}$ . Clearly, the right-hand side sums up to $1$ when $t=3$ .

Induction Hypothesis: Suppose the claim in Theorem˜10 is true for all graphs with less that $|V(G)|$ vertices.

Induction Step: If $G$ is disconnected, by the induction hypothesis, the statement of the theorem is true for each connected component and thus for $G$ . Therefore, assume $G$ to be connected. Let $k=max\{p(e)\mid e\in E(G)\}$ and $P$ be a $k$ -length path in $G$ . Let $P=v_{0},v_{1},\dots,v_{k}$ . Now consider the following two cases;

Case 1.

*There exists a $(k+1)$ -length cycle in $G$ .

Without loss of generality assume the endpoints of $P$ , $v_{0}$ and $v_{k}$ , are adjacent thus forming a $(k+1)$ -length cycle.*

Claim 1.

$V(P)=V(G)$ .

Proof.

We know that $v_{0}$ is adjacent to $v_{k}$ . Suppose $V(G)\setminus V(P)\neq\emptyset$ . Since $G$ is connected, there exists a vertex $v\in V(G)\setminus V(P)$ , adjacent to some vertex in $P$ . Since the vertices of $P$ form a cycle, without loss of generality, we can assume $v$ is adjacent to $v_{0}$ . Now consider the path;

[TABLE]

Thus $Q$ is a path in $G$ with length $k+1$ . Thus, we arrive at a contradiction to the assumption that the length of the longest path in $G$ is $k$ . ∎

From Claim˜1 we get that $p(e)=|V(G)|-1=n-1$ , for all $e\in E(G)$ . Therefore;

[TABLE]

Recall from the bound of Theorem˜9;

[TABLE]

Where $w(e)$ is the number of common neighbors of the endpoints of $e\in E(G)$ . Clearly $w(e)\leq n-2$ for all $e\in E(G)$ . Thus we get, $\frac{1}{{t\choose 2}}{w(e)\choose t-2}\leq\frac{1}{{t\choose 2}}{n-2\choose t-2}$ . Therefore from eq.˜13 we get that;

[TABLE]

Clearly, for equality $w(e)=n-2$ for all $e\in E(G)$ . Thus $G$ must be a clique.

Case 2.

*There does not exist any $(k+1)$ -length cycle in $G$ .

From the assumption of the case, $v_{0}$ and $v_{k}$ are non-adjacent. Without loss of generality assume $d(v_{0})\geq d(v_{k})$ .*

Claim 2.

$d(v_{k})\leq\frac{k}{2}$ **

Proof.

Suppose not, that is $d(v_{k})\geq\frac{k+1}{2}$ . Thus we get $d(v_{0})\geq\frac{k+1}{2}$ . Since $P$ is a longest path in $G$ and $v_{0}$ is not adjacent to $v_{k}$ , we get, $N(v_{0}),N(v_{k})\subseteq\{v_{1},v_{2},\dots,v_{k-1}\}$ . Let $N=N(v_{k})\setminus\{v_{k-1}\}$ and $N^{+}=\{v_{j+1}\mid v_{j}\in N\}$ . Clearly, $|N|=|N^{+}|\geq\frac{k+1}{2}-1=\frac{k-1}{2}$ . Note that $v_{0},v_{k}\notin N^{+}$ . Therefore, $|\{v_{1},v_{2},\dots,v_{k-1}\}\setminus N^{+}|\leq\frac{k-1}{2}$ . Since $|N(v_{0})|\geq\frac{k+1}{2}$ , by pigeonhole principle, we get that, there exists $v_{i}\in N^{+}$ such that $v_{i}\in N(v_{0})$ and $v_{i-1}\in N(v_{k})$ . Consider the cycle;

[TABLE]

Clearly $K$ is a $(k+1)$ -length cycle in $G$ . Thus, we get a contradiction. ∎

Define $G^{\prime}=G\setminus\{v_{k}\}$ . Now, using the induction hypothesis, we get that;

[TABLE]

where for $e\in E(G^{\prime})$ , $p_{G^{\prime}}(e)$ is the weight of the edge $e$ restricted to $G^{\prime}$ . Clearly $p_{G^{\prime}}(e)\leq p(e)$ for all $e\in E(G^{\prime})$ . Thus we get that;

[TABLE]

Note that the number of copies of $K_{t}$ containing the vertex $v_{k}$ is at most ${d(v_{k})\choose t-1}$ . Therefore,

[TABLE]

Claim 3.

$p(e)=k$ , for all $e\in E(G)\setminus E(G^{\prime})$ .

Proof.

Recall that, since $P$ is the longest path in $G$ , $N(v_{k})\subseteq\{v_{1},v_{2},\dots,v_{k-1}\}$ . Define $e_{i}=\{v_{i},v_{k}\}\in E(G)\setminus E(G^{\prime})$ where $v_{i}\in N(v_{k})$ . Thus $E(G)\setminus E(G^{\prime})=\{e_{i}\mid v_{i}\in N(v_{k})\}$ . Define $P_{i}$ to be the path formed by removing the edge $\{v_{i},v_{i+1}\}$ from $P$ and adding edge $e_{i}$ . Clearly, $P_{i}$ is also a $k$ -length path. Therefore, $p(e)=k$ for all $e\in E(G)\setminus E(G^{\prime})$ . ∎

Claim 4.

For $t\geq 3$ we have;

[TABLE]

Proof.

If ${d(v_{k})\choose t-1}=0$ , the bound is trivially true, since $p(e)\geq t-1$ for all $e\in E(G)$ . Thus we assume ${d(v_{k})\choose t-1}>0$ . From Claim˜3 we have that;

[TABLE]

Recall that from Claim˜2, $d(v_{k})\leq\frac{k}{2}$ . Therefore, we get;

[TABLE]

Since we assumed $t\geq 3$ , we get that;

[TABLE]

∎

From eqs.˜15 and 4 we get that;

[TABLE]

Clearly, we can not get equality in the bounds of Theorem˜10 for Case˜2

Equality in Theorem˜10 is possible only for Case˜1. Thus, we must have equality in eq.˜14 and therefore $w(e)=n-2$ for all $e\in E(G)$ ; thus, $G$ must be a clique. ∎

4 Concluding Remarks

In this paper, we developed localized versions of classical extremal bounds, focusing on the number of edges in planar graphs and clique bounds for graphs with bounded maximum degree and bounded path length. By introducing specifically designed local parameters, our results refine and extend the classical bound for planar graphs and improve the bounds of Wood [wood] and Chakraborty–Chen [CHAKRABORTI2024103955]. We also characterized the extremal graphs that attain these bounds. These results demonstrate the strength of the localization framework: by shifting the analysis from global to local parameters, it not only recovers known extremal results but also provides sharper bounds and new structural insights, underscoring its potential as a powerful tool for generalizing a wide range of extremal problems.

Statements and Declarations

Funding No funds, grants, or other support were received.

Data Availability No data was used for the research described in this article.

Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

TL;DR

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Localization: A Framework to Generalize Extremal Graph Problems

Abstract

keywords:

1 Introduction

Theorem 1**.**

Definition 1**.**

Theorem 2**.**

Notation 1**.**

Theorem 3**.**

Corollary 1**.**

Theorem 4**.**

Corollary 2**.**

Theorem 5**.**

1.1 Localization

Theorem 6**.**

2 Main Results

Notation 2**.**

Theorem 7**.**

Theorem 8**.**

Corollary 3**.**

Definition 2**.**

Notation 3**.**

Theorem 9**.**

Corollary 4**.**

Notation 4**.**

Theorem 10**.**

2.1 Recovering the classical bounds

3 Proof of Results

3.1 Proof of Theorem˜7

Proof.

3.2 Proof of Theorem˜8

Notation 5**.**

Lemma 1**.**

Proof.

Proof.

Claim 1**.**

Proof.

3.3 Proof of Theorem˜9

Lemma 2**.**

Proof.

Proof.

3.4 Proof of Theorem˜8 using Theorem˜9

Proof.

3.5 Proof of Theorem˜10

Lemma 3**.**

Proof.

Proof.

Case 1**.**

Claim 1**.**

Proof.

Case 2**.**

Claim 2**.**

Proof.

Claim 3**.**

Proof.

Claim 4**.**

Proof.

4 Concluding Remarks

Statements and Declarations

References

Theorem 1.

Definition 1.

Theorem 2.

Notation 1.

Theorem 3.

Corollary 1.

Theorem 4.

Corollary 2.

Theorem 5.

Theorem 6.

Notation 2.

Theorem 7.

Theorem 8.

Corollary 3.

Definition 2.

Notation 3.

Theorem 9.

Corollary 4.

Notation 4.

Theorem 10.

Notation 5.

Lemma 1.

Claim 1.

Lemma 2.

Lemma 3.

Case 1.

Claim 1.

Case 2.

Claim 2.

Claim 3.

Claim 4.