On the maximum order of graphs embedded in surfaces

Eran Nevo, Guillermo Pineda-Villavicencio, David R. Wood

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

The maximum number of vertices in a graph of maximum degree δ≥3 and fixed diameter k≥2 is upper bounded by (1+o(1))(δ-1)k. If we restrict our graphs to certain classes, better upper bounds are known. For instance, for the class of trees there is an upper bound of (2+o(1))(δ-1)⌊k/2⌋ for a fixed k. The main result of this paper is that graphs embedded in surfaces of bounded Euler genus g behave like trees, in the sense that, for large δ, such graphs have orders bounded from above by(c(g+1)(δ-1)⌊k/2⌋if k is evenc(g3/2+1)(δ-1)⌊k/2⌋if k is odd, where c is an absolute constant. This result represents a qualitative improvement over all previous results, even for planar graphs of odd diameter k. With respect to lower bounds, we construct graphs of Euler genus g, odd diameter k, and order c(g+1)(δ-1)⌊k/2⌋ for some absolute constant c>0. Our results answer in the negative a question of Miller and Širáň (2005).

Original languageAmerican English
Pages (from-to)28-41
Number of pages14
JournalJournal of Combinatorial Theory. Series B
Volume119
DOIs
StatePublished - 1 Jul 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2016 Elsevier Inc.

Keywords

  • Degree-diameter problem
  • Graphs on surfaces
  • Surface
  • Vertex separator

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