Criteria on forbidden subgraphs in the complements for positive Lin--Lu--Yau curvature

TL;DR

This paper establishes conditions on forbidden subgraphs in graph complements that ensure positive Lin–Lu–Yau curvature, providing new bounds and constructions for specific subgraph exclusions.

Contribution

It proves new criteria linking forbidden subgraphs in complements to positive curvature, including optimal bounds and counterexamples for various subgraph types.

Findings

Graphs with complement no 4-cycle (except 4-vertex path) have positive curvature.

Graphs with complement no K_{2,t} and sufficiently many vertices have positive curvature.

The established bounds are optimal for t ≥ 10.

Abstract

We investigate forbidden subgraph conditions in the complement of a graph that guarantee positive Lin--Lu--Yau curvature. In particular, we prove that every graph whose complement contains no $4$-cycles has positive Lin--Lu--Yau curvature, with the only exception of the $4$-vertex path. We further prove that, for any integer $t\ge2$, every graph on at least $\max\{t^2-2t+2, 8t\}$ vertices whose complement contains no $K_{2,t}$ has positive curvature. In addition, this lower bound on the number of vertices is optimal for $t\geq 10$. Finally, we construct examples showing that, in general, the forbidden subgraphs in these results cannot be replaced by cycles of length other than $4$ or by complete bipartite graphs $K_{s,t}$ with $s> 2$ and $t> 2$.