Subgraphs of graphs on surfaces with high representativity

Ken Ichi Kawarabayashi, Atsuhiro Nakamoto, Katsuhiro Ota

Research output: Contribution to journalArticle

13 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)207-229
Number of pages23
JournalJournal of Combinatorial Theory. Series B
Volume89
Issue number2
DOIs
Publication statusPublished - 2003 Nov

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Graphs on Surfaces
Connected graph
Subgraph
Covering
Degree Sum
Klein bottle
Vertex Degree
Sharp Bound
Bottles
Projective plane
Walk
Euler
Torus
Genus
Closed
Integer
Graph in graph theory

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Subgraphs of graphs on surfaces with high representativity. / Kawarabayashi, Ken Ichi; Nakamoto, Atsuhiro; Ota, Katsuhiro.

In: Journal of Combinatorial Theory. Series B, Vol. 89, No. 2, 11.2003, p. 207-229.

Research output: Contribution to journalArticle

Kawarabayashi, Ken Ichi ; Nakamoto, Atsuhiro ; Ota, Katsuhiro. / Subgraphs of graphs on surfaces with high representativity. In: Journal of Combinatorial Theory. Series B. 2003 ; Vol. 89, No. 2. pp. 207-229.
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