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

  • Authors:

  • S. Jendrol;H.-J. Voss

  • Affiliations:

  • Institute of Mathematics, Faculty of Science, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovakia;Department of Algebra, Technical University Dresden, Mommsenstrasse 13, D-01062 Dresden, Germany

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2005

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Abstract

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) 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 satisfies ΔG(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. 2. Each graph G ∈ g(M) of order at least k ≥ 5 contains a connected subgraph H of order k such that its maximum degree ΔG(H) satisfies ΔG(H) ≤ 2 + ⌊(4k - 2)(1 + |χ(M)|/3)⌋. This bound is best possible.The sharp values of ΔG(Pk) and ΔG(H) are determined for k ∈ {2, 3, 4} as well.