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 (M) 0. We prove that each polyhedral map G on M with minimum degree and large number of vertices contains a k-path P, a path on k vertices, such that: (i) for 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 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 = 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 o...

• Type: Journal
• Year: 2002
• Where: ENDM
• Authors: Stanislav Jendrol, Heinz-Jürgen Voss
• Keywords: ENDM 2002, Euler Characteristic, Minimum Degree, Polyhedral Maps