Light paths in large polyhedral maps with prescribed minimum degree

Let k be an integer and M be a closed 2-manifold with Euler characteristic χ(M) ≤ 0. We prove that each polyhedral map G on M with minimum degree δ and large number of vertices contains a k-path P , a path on k vertices, such that: (i) for δ ≥ 4 every vertex of P has, in G, degree bounded from above by 6k − 12, k ≥ 8 (It is also shown that this bound is tight for k even and that for k odd this bound cannot be lowered below 6k − 14); (ii) for δ ≥ 5 and k ≥ 68 every vertex of P has, in G, a degree bounded from above by 6k−2 log2 k+2. For every k ≥ 68 and for every M we construct a large polyhedral map such that each k-path in it has a vertex of degree at least 6k − 72 log2(k − 1) + 112. (iii) The case δ = 3 was dealt with in an earlier paper of the authors (Light paths with an odd number of vertices in large polyhedral maps. Annals of Combinatorics 2(1998), 313-324) where it is shown that every vertex of P has, in G, a degree bounded from above by 6k if k = 1 or k even, and by 6k − 2 if k ≥ 3, k odd; these bounds are sharp. The paper also surveys previous results in this field. Australasian Journal of Combinatorics 25(2002), pp.79–102