The Edge Spectrum of $$K_4^-$$ 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 the 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 $$n\ge 8$$ 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 (Graphs Combin 34:85–95, 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.