Subgraphs of graphs on surfaces with high representativity

  • Authors:

  • Ken-ichi Kawarabayashi;Atsuhiro Nakamoto;Katsuhiro Ota

  • Affiliations:

  • Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ;Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan;Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2003

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Abstract

Let G be a 3-connected graph with n vertices on a non-spherical closed surface Fk2 of Euler genus k with sufficiently large representativity. In this paper, we first study a new cutting method which produces a spanning planar subgraph of G with a certain good property. This is used to show that such a graph G has a spanning 4-tree with at most max{2k - 5, 0} vertices of degree 4. Using this result, we prove that for any integer t, if n is sufficiently large, then G has a connected subgraph with t vertices whose degree sum is at most 8t - 1. We also give a nearly sharp bound for the projective plane, torus and Klein bottle. Furthermore, using our cutting method, we prove that a 3-connected graph G on Fk2 with high representativity has a 3-walk in which at most max{2k - 4, 0} vertices are visited three times, and an 8-covering with at most max{4k - 8, 0} vertices of degree 7 or 8. Moreover, a 4-connected G has a 4-covering with at most max{4k - 6, 0} vertices of degree 4.