Light and low 5-stars in normal plane maps with minimum degree 5

Abstract

It is known that there are normal plane maps (NPMs) with minimum degree δ = 5 such that the minimum degree-sum w(S₅) of 5-stars at 5-vertices is arbitrarily large. The height of a 5-star is the maximum degree of its vertices. Given an NPM with δ = 5, by h(S₅) we denote the minimum height of a 5-stars at 5-vertices in it.

Lebesgue showed in 1940 that if an NPM with δ = 5 has no 4-stars of cyclic type \(\overrightarrow {\left( {5,6,6,5} \right)} \) centered at 5-vertices, then w(S₅) ≤ 68 and h(S₅) = 41. Recently, Borodin, Ivanova, and Jensen lowered these bounds to 55 and 28, respectively, and gave a construction of a \(\overrightarrow {\left( {5,6,6,5} \right)} \)-free NPM with δ = 5 having w(S₅) = 48 and h(S₅) = 20.

In this paper, we prove that w(S₅) ≤ 51 and h(S₅) ≤ 23 for each (\(\overrightarrow {\left( {5,6,6,5} \right)} \)-free NPM with δ = 5.