Shortness parameters of polyhedral graphs with few distinct vertex degrees

Abstract

We devise several new upper bounds for shortness parameters of regular polyhedra and of the polyhedra that have two vertex degrees, and relate these to each other. Grünbaum and Walther showed that quartic polyhedra have shortness exponent at most $\log 22/\log 23$. This was subsequently improved by Harant to $\log 16/\log 17$, which holds even when all faces are either triangles or of length k, for infinitely many k. We complement Harant's result by strengthening the Grünbaum-Walther bound to $\log 4/\log 5$, and showing that this bound even holds for the family of quartic polyhedra with faces of length at most 7. Furthermore, we prove that for every $4\le \ell \le 8$ the shortness exponent of the polyhedra having only vertices of degree 3 or ℓ is at most $\log 5/\log 7$. Motivated by work of Ewald, we show that polyhedral quadrangulations with maximum degree at most 5 have shortness coefficient at most 30/37. Finally, we define path analogues for shortness parameters, and propose first dependencies between these measures.