### Every Triangulated 3-Polytope of Minimum Degree 4 has a 4-Path of Weight at Most 27

#### Abstract

By $\delta$ and $w_k$ denote the minimum degree and minimum degree-sum (weight) of a $k$-vertex path in a given graph, respectively. For every 3-polytope, $w_2\le13$ (Kotzig, 1955) and $w_3\le21$ (Ando, Iwasaki, Kaneko, 1993), where both bounds are sharp. For every 3-polytope with $\delta\ge4$, we have sharp bounds $w_2\le11$ (Lebesgue, 1940) and $w_3\le17$ (Borodin, 1997).

Madaras (2000) proved that every triangulated 3-polytope with $\delta\ge4$ satisfies $w_4\le31$ and constructed such a 3-polytope with $w_4=27$.

We improve the Madaras bound $w_4\le31$ to the sharp bound $w_4\le27$.

#### Keywords

Plane graph; Structural property; Normal plane map; 4-Path