Light 3-stars in embedded graphs

By g(k,t) we denote the smallest integer such that every plane graph with girth g≥g(k,t), minimum degree at least 2, and no (k+1)-paths consisting of vertices of degree 2, has a 3-vertex with at least t neighbors of degree 2. There are many results concerning bounds on g(k,t) for planar graphs. Borodin and Ivanova completed the list of values of g(k,t) for all k≥1 and 1≤t≤3. In the case of graphs embedded on surfaces of higher genus we need to consider additional condition on the number of vertices of these graphs (depending on the Euler characteristic of given graph). By g⁎(k,t) we denote the smallest integer for which there exists a function f(k,t) such that for every surface S with non-positive Euler characteristic χ(S) and every graph G embedded on S with |V(G)|>f(k,t)|χ(S)|, g(G)≥g⁎(k,t), δ(G)≥2, and no (k+1)-path consisting of vertices of degree 2, k≥1, G has a 3-vertex with at least t neighbors of degree 2. In this paper we prove that g⁎(k,1)=4k+5 for f(k,1)≥10k+5, g⁎(k,2)=4k+5 for f(k,2)≥max⁡{10k+5,8k+15+204k+1}, and g⁎(k,3)=4k+7 for f(k,3)≥30k+30. Moreover, we will discuss the quality of our results.