Sparse Graphic Degree Sequences Have Planar Realizations

A sequence d = (d₁, d₂, . . ., d_n) 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 ∑ d ≤ 2n − 2, then d has a realization by a forest, hence it is trivially planaric. In this paper, we seek bounds on ∑ d that guarantee that if d is graphic then it is also planaric. We show that this holds true when ∑ d ≤ 4n − 4 − 2ω₁, where ω₁ is the number of 1’s in d. Conversely, we show that there are graphic sequences with ∑ d = 4n − 2ω₁ that are non-planaric. For the case ω₁ = 0, we show that d is planaric when ∑ d ≤ 4n − 4. Conversely, we show that there is a graphic sequence with ∑ d = 4n − 2 that is non-planaric. In fact, when ∑ d ≤ 4n − 6 − 2ω₁, d can be realized by a graph with a 2-page book embedding.