On the least size of a graph with a given degree set -- II (2009.10294v1)

Abstract: The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $\mathscr D$ is $1+\max \mathscr D$. Tripathi & Vijay considered the analogous problem concerning the least size of graphs with degree set $\mathscr D$. We expand on their results, and determine the least size of graphs with degree set $\mathscr D$ when (i) $\min \mathscr D \mid d$ for each $d \in \mathscr D$; (ii) $\min \mathscr D=2$; (iii) $\mathscr D={m,m+1,\ldots,n}$. In addition, given any $\mathscr D$, we produce a graph $G$ whose size is within $\min \mathscr D$ of the optimal size, giving a $\big(1+\frac{2}{d_1+1})$-approximation, where $d_1=\max \mathscr D$.