Light classes of generalized stars in polyhedral maps on surfaces

A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. Si denotes a generalized 3-star, all three maximal paths starting in Z have exactly i + 1 vertices (including Z). Let M be a surface of Euler characteristic χ(M) ≤ 0, and m(M) := b 5+ √ 49−24χ(M) 2 c. We prove: (1) Let k ≥ 1, d ≥ m(M) be integers. Each polyhedral map G on M with a k-path (on k vertices) contains a k-path of maximum degree ≤ d in G or a generalized s-star T, s ≤ m(M), on d+2−m(M) vertices with root Z, where Z has degree ≤ k ·m(M) and the maximum degree of T r {Z} is ≤ d in G. Similar results are obtained for the plane and for large polyhedral maps on M. 86 S. Jendrol’ and H.-J. Voss (2) Let k and i be integers with k ≥ 3, 1 ≤ i ≤ k2 . If a polyhedral map G on M with a large enough number of vertices contains a k-path then G contains a k-path or a 3-star Si of maximum degree ≤ 4(k + i) in G. This bound is tight. Similar results hold for plane graphs.