The edge spectrum of $K_4^-$-saturated graphs

TL;DR

This paper characterizes the possible sizes of $K_4^-$-saturated graphs, showing conditions under which such graphs are bipartite or non-bipartite, and completes the determination of their edge spectrum.

Contribution

It fully determines the edge spectrum of $K_4^-$-saturated graphs, resolving a conjecture and extending previous results.

Findings

Bipartite $K_4^-$-saturated graphs have size exceeding a specific bound.

Existence of non-bipartite $K_4^-$-saturated graphs within a certain size interval.

Complete characterization of the edge spectrum of $K_4^-$-saturated graphs.

Abstract

Given graphs $G$ and $H$, $G$ is $H$-saturated if $G$ does not contain a copy of $H$ but the addition of any edge $e\notin E(G)$ creates at least one copy of $H$ within $G$. The edge spectrum of $H$ is the set of all possible sizes of an $H$-saturated graph on $n$ vertices. Let $K_4^-$ be a graph obtained from $K_4$ by deleting an edge. In this note, we show that (a) if $G$ is a $K_4^-$-saturated graph with $|V(G)|=n$ and $|E(G)|>\lfloor \frac{n-1}{2} \rfloor \lceil \frac{n-1}{2} \rceil +2$, then $G$ must be a bipartite graph; (b) there exists a $K_4^-$-saturated non-bipartite graph on $n\ge 10$ vertices with size being in the interval $\left[3n-11, \left\lfloor \frac{n-1}{2} \right\rfloor \left\lceil \frac{n-1}{2} \right\rceil +2\right]$. Together with a result of Fuller and Gould in [{\it On ($\hbox{K}_t-e$)-Saturated Graphs. Graphs Combin., 2018}], we determine the edge spectrum of $K_4^-$ completely, and a conjecture proposed by Fuller and Gould in the same paper also has been resolved.