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) ≥ 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); ≥ 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.); uthors have proved in their previous papers that) for δ = 3 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. per also surveys previous results in this field.