Graphs and Combinatorics

, Volume 29, Issue 1, pp 105–119

On Superconnectivity of (4, g)-Cages

  • Hongliang Lu
  • Yunjian Wu
  • Yuqing Lin
  • Qinglin Yu
  • Camino Balbuena
  • Xavier Marcote
Original Paper

DOI: 10.1007/s00373-011-1091-5

Cite this article as:
Lu, H., Wu, Y., Lin, Y. et al. Graphs and Combinatorics (2013) 29: 105. doi:10.1007/s00373-011-1091-5
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Abstract

A (k, g)-cage is a graph that has the least number of vertices among all k-regular graphs with girth g. It has been conjectured (Fu et al. in J. Graph Theory, 24:187–191, 1997) that all (k, g)-cages are k-connected for every k ≥ 3. A k-connected graph G is called superconnected if every k-cutset S is the neighborhood of some vertex. Moreover, if GS has precisely two components, then G is called tightly superconnected. In this paper, we prove that every (4, g)-cage is tightly superconnected when g ≥ 11 is odd.

Keywords

CageSuperconnectedTightly superconnected

Copyright information

© Springer 2011

Authors and Affiliations

  • Hongliang Lu
    • 1
  • Yunjian Wu
    • 2
  • Yuqing Lin
    • 3
  • Qinglin Yu
    • 4
  • Camino Balbuena
    • 5
  • Xavier Marcote
    • 5
  1. 1.Department of MathematicsXi’an Jiaotong UniversityXi’anPeople’s Republic of China
  2. 2.Department of MathematicsSoutheast UniversityNanjingPeople’s Republic of China
  3. 3.School of Electrical Engineering and Computer ScienceThe University of NewcastleNewcastleAustralia
  4. 4.Department of Mathematics and StatisticsThompson Rivers UniversityKamloopsCanada
  5. 5.Departament de Matemàtica Aplicada IIIUniversitat Politècnica de CatalunyaBarcelonaSpain