Vertex-Based Localization of ErdH{o}s-Gallai Theorems for Paths and Cycles

2025-04-02

AI Summary

This paper addresses the localization problem of the Erdős–Gallai theorem by introducing the first vertex-level generalization: replacing the global path and cycle lengths in the original theorem with vertex-specific parameters—the length (p(v)) of a longest path through vertex (v), and the length (c(v)) of a longest cycle containing (v). Methodologically, the work integrates combinatorial analysis, structural induction, and extremal construction. It establishes two tight upper bounds on the number of edges (m): (m leq frac{1}{2}sum_v p(v)) for path-free graphs, and (m leq frac{1}{2}sum_v c(v) - frac{1}{2}c(u)) for certain cycle-containing graphs, where (u) is a distinguished vertex. Crucially, the paper fully characterizes the extremal graphs achieving equality in both bounds—providing the first complete structural description of such graphs. This advances the theory from global constraints to fine-grained, per-vertex contributions, markedly enhancing the capacity to analyze local structure in sparse graphs.

Technology Category

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

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: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The ErdH{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m leq frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that [m leq sum_{v in V(G)} frac{p(v)}{2}] The ErdH{o}s-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the longest cycle) at most $k$, we have $m leq frac{k(n-1)}{2}$. We strengthen this result as follows: 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 [m leq left( sum_{v in V(G)} frac{c(v)}{2} ight) - frac{c(u)}{2}] where $c(u)$ denotes the circumference of $G$. ewline Furthermore, we characterize the class of extremal graphs that attain equality in these bounds.

Problem

• Generalizes Erdős-Gallai theorem for paths using vertex-based longest paths
• Strengthens Erdős-Gallai theorem for cycles with vertex-based cycle lengths
• Characterizes extremal graphs achieving equality in derived bounds

Innovation

• Vertex-based generalization of path bounds
• Enhanced cycle bounds using vertex data
• Characterization of extremal graph equality