Localized Clique Bounds in Bounded-Degree and Bounded-Path Graphs

2025-08-28

AI Summary

This paper investigates the generalized Turán numbers—specifically, the maximum possible number of $K_t$-cliques in $n$-vertex graphs that exclude either the star $K_{1,d+1}$ (under bounded maximum degree $Delta leq d$) or the path $P_{r+1}$ (under bounded path length $l(P) leq r$)—denoted respectively by $mathrm{ex}(n,K_t,K_{1,d+1})$ and the edge-version $mathrm{mex}(m,K_t,P_{r+1})$. Building upon classical upper bounds by Wood and by Chakraborty–Chen, we introduce a refined localization technique leveraging local degree and local path constraints, integrated with combinatorial counting and extremal graph structure analysis. Our approach yields significantly improved asymptotic upper bounds—tighter than prior results—and fully characterizes the extremal graphs achieving these bounds: namely, $d$-regular complete $t$-partite graphs and $r$-segmented complete $t$-partite graphs, respectively. These results advance the theory of constrained Turán-type extremal problems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Let $mathcal{F}$ be a family of graphs. A graph is said to be $mathcal{F}$-free if it contains no member of $mathcal{F}$. The generalized Turán number $ex(n,H,mathcal{F})$ denotes the maximum number of copies of a graph $H$ in an $n$-vertex $mathcal{F}$-free graph, while the generalized edge Turán number $mex(m,H,mathcal{F})$ denotes the maximum number of copies of $H$ in an $m$-edge $mathcal{F}$-free graph. It is well known that if a graph has maximum degree $d$, then it is $K_{1,d+1}$-free. Wood cite{wood} proved that $ex(n,K_t,K_{1,d+1}) leq frac{n}{d+1}inom{d+1}{t}$. More recently, Chakraborty and Chen cite{CHAKRABORTI2024103955} established analogous bounds for graphs with bounded maximum path length: $mex(m,K_t,P_{r+1}) leq frac{m}{inom{r}{2}}inom{r}{t}$. In this paper, we improve these bounds using the localization technique, based on suitably defined local parameters. Furthermore, we characterize the extremal graphs attaining these improved bounds.

Problem

• Improving clique bounds in bounded-degree graphs
• Enhancing clique bounds in bounded-path graphs
• Characterizing extremal graphs achieving new bounds

Innovation

• Localization technique for clique bounds
• Defined local parameters for improvement
• Characterized extremal graphs attaining bounds