Sparse Graphic Degree Sequences Have Planar Realizations

Abstract

A sequence $d=\left(d_1,d_2,\dots ,d_n\right)$ of positive integers is graphic if it is the degree sequence of some simple graph $G$, and planaric if it is the degree sequence of some simple planar graph $G$. It is known that if $\sum d\le 2n-2$, then $d$ has a realization by a forest, hence it is trivially planaric. This condition for $d$ to be realized by a forest is tight, since clearly $\sum d$ must be even, and it is known that if $\sum d=2n$ then $d$ does not have a forest realization. In this paper, we seek bounds on $\sum d$ that guarantee that if $d$ is graphic then it is also planaric. We show that this holds true when $\sum d\le 4n-4-2{\omega }_1$, where ${\omega }_1$ is the number of 1's in $d$. Conversely, we show that there are graphic sequences with $\sum d=4n-2{\omega }_1$ that are non-planaric. For the case ${\omega }_1=0$, we show that $d$ is planaric when $\sum d\le 4n-4$. Conversely, we show that there is a graphic sequence with $\sum d=4n-2$ that is non-planaric. In fact, when $\sum d\le 4n-6-2{\omega }_1,d$ can be realized by a graph with a 2-page book embedding.