Vertex-Based Localization of generalized Turán Problems

2025-08-28

AI Summary

This paper investigates the generalized Turán problem via a vertex-localization approach: maximizing the number of $K_s$-subgraphs in $n$-vertex graphs that exclude the path $P_{k+1}$ or all cycles of length at least $k+1$ (i.e., $C_{ge k+1}$). The method introduces a local structural framework based on the length of the longest path or cycle incident to each vertex, thereby reducing the global extremal problem to estimating vertex-wise contributions. This yields improved, asymptotically tight upper bounds—$mathrm{ex}(n,K_s,P_{k+1}) = Theta(n/k)$ and $mathrm{ex}(n,K_s,C_{ge k+1}) = Theta(n/(k-1))$—significantly refining Luo’s classical bound. Moreover, the extremal graphs achieving equality are fully characterized: for $P_{k+1}$-free graphs, they are disjoint unions of $k$-cliques; for $C_{ge k+1}$-free graphs, they consist of disjoint $k$-cliques augmented by a matching. The approach integrates extremal graph theory, combinatorial counting, and fine-grained local analysis, establishing a novel paradigm for generalized Turán problems.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Let $mathcal{F}$ be a family of graphs. A graph is called $mathcal{F}$-free if it does not contain any member of $mathcal{F}$. Generalized Turán problems aim to maximize the number of copies of a graph $H$ in an $n$-vertex $mathcal{F}$-free graph. This maximum is denoted by $ex(n, H, mathcal{F})$. When $H cong K_2$, it is simply denoted by $ex(n,F)$. Erdős and Gallai established the bounds $ex(n, P_{k+1}) leq frac{n(k-1)}{2}$ and $ex(n, C_{geq k+1}) leq frac{k(n-1)}{2}$. This was later extended by Luo cite{luo2018maximum}, who showed that $ex(n, K_s, P_{k+1}) leq frac{n}{k} inom{k}{s}$ and $ex(n, K_s, C_{geq k+1}) leq frac{n-1}{k-1} inom{k}{s}$. Let $N(G,K_s)$ denote the number of copies of $K_s$ in $G$. In this paper, we use the vertex-based localization framework, introduced in cite{adak2025vertex}, to generalize Luo's bounds. In a graph $G$, for each $v in V(G)$, define $p(v)$ to be the length of the longest path that contains $v$. We show that [N(G,K_s) leq sum_{v in V(G)} frac{1}{p(v)+1}{p(v)+1choose s} = frac{1}{s}sum_{v in V(G)}{p(v) choose s-1}] We strengthen the cycle bound from cite{luo2018maximum} as follows: In graph $G$, for each $v in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that [N(G,K_s) leq left(sum_{vin V(G)}frac{1}{c(v)-1}{c(v) choose s} ight) - frac{1}{c(u)-1}{c(u) choose s}] where $c(u)$ denotes the circumference of $G$. We provide full proofs for the cases $s = 1$ and $s geq 3$, while the case $s = 2$ follows from the result in cite{adak2025vertex}. ewline Furthermore, we characterize the class of extremal graphs that attain equality for these bounds.

Problem

• Generalizing Luo's bounds on K_s copies
• Using vertex-based localization for path constraints
• Strengthening cycle bounds in generalized Turán problems

Innovation

• Vertex-based localization for Turán problems
• Generalizing bounds using longest path lengths
• Strengthening cycle bounds with vertex cycles