Calculating the extremal number ex(v;{C3,C4,...,Cn})

  • Authors:

  • Jianmin Tang;Yuqing Lin;Camino Balbuena;Mirka Miller

  • Affiliations:

  • School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, Victoria 3353, Australia;School of Electrical Engineering and Computer Science, The University of Newcastle, NSW 2308, Australia;Departament de Matemítica Aplicada III, Universitat Politècnica de Catalunya, Campus Nord, Edifici C2, C/ Jordi Girona 1 i 3, E-08034 Barcelona, Spain;School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, Victoria 3353, Australia and Department of Mathematics, University of West Bohemia, Pilsen, Czech Repu ...

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

By the extremal numberex(v;{C"3,C"4,...,C"n}) we denote the maximum number of edges in a graph of order v and girth at least g=n+1. The set of such graphs is denoted by EX(v;{C"3,C"4,...,C"n}). In 1975, Erdos mentioned the problem of determining extremal numbers ex(v;{C"3,C"4}) in a graph of order v and girth at least five. In this paper, we consider a generalized version of the problem for any value of girth by using the hybrid simulated annealing and genetic algorithm (HSAGA). Using this algorithm, some new results for n=5 have been obtained. In particular, we generate some graphs of girth 6,7 and 8 which in some cases have more edges than corresponding cages. Furthermore, future work will be described regarding the investigation of structural properties of such extremal graphs and the implementation of HSAGA using parallel computing.