Long cycles in graphs on a fixed surface

  • Authors:

  • Thomas Böhme;Bojan Mohar;Carsten Thomassen

  • Affiliations:

  • Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany;Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia;Department of Mathematics, Technical University of Denmark, Lyngby, Denmark

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2002

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Abstract

We prove that there exists a function a: N0 × R+ → N such that (i) If G is a 4-connected graph of order n embedded on a surface of Euler genus g such that the face-width of G is at least a(g, ε), then G can be covered by two cycles each of which has length at least (1-ε) n. We apply this to derive lower bounds for the length of a longest cycle in a graph G on any fixed surface. Specifically, there exist functions b: N0 → N and c: N0 → R+ such that for every graph G on n vertices that is embedded on a surface of Euler genus g the following statements hold: (ii) If G is 4-connected, then G contains a collection of at most b(g) paths which cover all vertices of G, and G contains a cycle of length at least n/b(g). (iii) If G is 3-connected, then G contains a cycle of length at least c(g)nlog2/log3. Moreover, for each ε 0, every 4-connected graph G with sufficiently large face-width contains a spanning tree of maximmn degree at most 3 and a 2-connected spanning subgraph of maximum degree at most 4 such that the number of vertices of degree 3 or 4 in either of these subgraphs is at most ε |V(G)|.