Light and low 5-stars in normal plane maps with minimum degree 5
Oleg Veniaminovich Borodin, Anna O. Ivanova · Siberian Mathematical Journal · 2016
It is known that there are normal plane maps (NPMs) with minimum degree δ = 5 such that the minimum degree-sum w(S 5) 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(S 5) 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(S 5) ≤ 68 and h(S 5) = 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(S 5) = 48 and h(S 5) = 20. In this paper, we prove that w(S 5) ≤ 51 and h(S 5) ≤ 23 for each ( $$\overrightarrow {\left( {5,6,6,5} \right)} $$ -free NPM with δ = 5.