Toward Grünbaum’s conjecture bounding vertices of degree 4

Abstract

Given a spanning tree $T$ of a planar graph $G$, the co-tree of $T$ is the spanning tree of the dual graph $G^{*}$ with edge set ${(E\left(G\right)-E\left(T\right))}^{*}$. Grünbaum conjectured in 1970 that every planar 3-connected graph $G$ contains a spanning tree $T$ such that both $T$ and its co-tree have maximum degree at most 3.

While Grünbaum’s conjecture remains open, Schmidt and the author recently improved the upper bound on the maximum degree from 5 (Biedl 2014) to 4.

In this paper, we modify this approach taking a further step towards Grünbaum’s conjecture. We again obtain a spanning tree $T$ such that both $T$ and its co-tree have maximum degree at most 4 and, additionally, an upper bound on the number of vertices of degree 4 of $T$ and its co-tree.