LIGHT 3-STARS IN SPARSE PLANE GRAPHS
A k-star S<sub>k</sub>(v) in a plane graph G consists of a central vertex v and k its neighbor vertices. The height h(S<sub>k</sub> (v)) and weight w(S<sub>k</sub>(v)) of Sfc(v) is the maximum degree and degree-sum of its vertices, respectively. The height h<sub>k</sub>(G) and weight w<sub>k</sub>(G) of G is the maximum height and weight of its k-stars. Lebesgue (1940) proved that every 3-polytope of girth g at least 5 has a 2-star (a path of three vertices) with h<sub>2</sub> = 3 and w<sub>2</sub> = 9. Madaras (2004) refined this by showing that there is a 3-star with h<sub>3</sub> = 4 and w<sub>3</sub> = 13, which is tight. In 2015, we gave another tight description of 3-stars for girth g = 5 in terms of degree of their vertices and showed that there are only these two tight descriptions of 3-stars. In 2013, we gave a tight description of 3<sup>-</sup>-stars in arbitrary plane graphs with minimum degree 5 at least 3 and g > 3, which extends or strengthens several previously known results by Balogh, Jendrol', Harant, Kochol, Madaras, Van den Heuvel, Yu and others and disproves a conjecture by Harant and Jendrol’ posed in 2007. There exist many tight results on the height, weight and structure of 2<sup>-</sup>-stars when 5 = 2. In 2016, Hudak, Macekova, Madaras, and Siroczki considered the class of plane graphs with 5 = 2 in which no two vertices of degree 2 are adjacent. They proved that h<sub>3</sub> = w<sub>3</sub> = to if g < 6, h<sub>3</sub> = 5 if g = 7, h<sub>3</sub> = 3 if g > 8, w<sub>3</sub> = 10 if g = 8 and w<sub>3</sub> = 3 if g > 9. For g = 7, Hudak et al. proved 11 < w<sub>3</sub> < 20. The purpose of our paper is to prove that every plane graph with 5 = 2, g = 7 and no adjacent vertices of degree 2 has w<sub>3</sub> = 12.