Work

A sharp bound on the binding number of $K_k$-free graphs and the Brandt conjecture

Year: 2026

Type: preprint

Abstract: The binding number of a graph is the smallest ratio $|N(S)|/|S|$ among all nonempty vertex sets $S$ whose neighborhood $N(S)$ is not the whole vertex set. We prove that every finite nonempty $K_k$-free graph has binding number at most $k-2$ for every $k\ge6$, confirming Brandt's conjecture; the bound is sharp for complete $(k-1)$-partite graphs with equal nonempty parts. We first prove the case $k=6$ and then extend it to all $k\ge6$ by induction.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28536

Authors: Lin, Jifu, Yang, Heng

Institution: East China Normal University

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7223116079

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