Hua–Peng problem on the saturation number of the kite graph

Let $K$ be the kite graph obtained from $K_4$ by deleting one edge and then attaching a pendant edge to one of the two vertices of degree $2$ in the resulting graph. For every integer $n\ge 5$ , determine the saturation number $\operatorname{sat}(n,K)$ , where $\operatorname{sat}(n,K)$ is the minimum number of edges in a $K$ -saturated graph on $n$ vertices, and characterize all $K$ -saturated graphs on $n$ vertices attaining this minimum.

Progress summary

Solved

A new paper settles the kite graph’s minimum-edge saturation problem and classifies all graphs achieving it, while the larger Hua–Peng program remains only partly answered.

Hua and Peng’s 2026 problem asks for the saturation number of the kite graph and the classification of all extremal saturated graphs. Their broader problem concerns the remaining undetermined connected graphs on five vertices; the kite is one of three cases.

August 2026 resolution

Bian, Huanying; Cui, Qing; Ji, Shengjin; and Ma, Fufong determine $\operatorname{sat}(n,K)$ for every $n \ge 5$ and characterize all extremal $K$ -saturated graphs as the family $\mathcal{K}_n$ , including disconnected cases. The arXiv paper supplies proofs and reports no unresolved objection; it is a partial answer only to Hua and Peng’s broader program.

Current status (as of August 2026): The kite-graph case is settled by an arXiv paper, including its saturation formula and all extremal graphs; the other cases in Hua and Peng’s broader problem remain open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

On the saturation number of the kite graph — arXiv — Bian, Huanying, Cui, Qing, Ji, Shengjin, Ma, Fufong

extremal-graph-theory

2 days ago