Two local and one global properties of 3-connected graphs on compact 2-dimensional manifolds

Let G(M) be the family of all 3-connected graphs which can be embedded in a compact 2-manifold M with Euler characteristic χ(M)<0. We have proved the following two results: 1. raph G∈G(M) having a k-path, a path on k-vertices, k⩾4, contains a k-path Pk such that its maximum degree ΔG(Pk) in G satisfiesΔG(Pk)⩽2+(6k−6−2ϵ)1+|χ(M)|3,where ϵ=0 for even k and ϵ=1 for odd k. This bound is best possible. raph G∈G(M) of order at least k⩾5 contains a connected subgraph H of order k such that its maximum degree ΔG(H) satisfiesΔG(H)⩽2+(4k−2)1+|χ(M)|3.This bound is best possible. arp values of ΔG(Pk) and ΔG(H) are determined for k∈{2,3,4} as well.