Original Paper

Graphs and Combinatorics

, Volume 29, Issue 1, pp 105-119

On Superconnectivity of (4, g)-Cages

  • Hongliang LuAffiliated withDepartment of Mathematics, Xi’an Jiaotong University Email author 
  • , Yunjian WuAffiliated withDepartment of Mathematics, Southeast University
  • , Yuqing LinAffiliated withSchool of Electrical Engineering and Computer Science, The University of Newcastle
  • , Qinglin YuAffiliated withDepartment of Mathematics and Statistics, Thompson Rivers University
  • , Camino BalbuenaAffiliated withDepartament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya
  • , Xavier MarcoteAffiliated withDepartament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya

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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

Cage Superconnected Tightly superconnected