New families of graphs without short cycles and large size

  • Authors:

  • E. Abajo;C. Balbuena;A. Diánez

  • Affiliations:

  • Departamento de Matemática Aplicada I, Universidad de Sevilla, Avda Reina Mercedes s/n, E-41012 Sevilla, Spain;Departament de Matemítica Aplicada III, Universitat Politècnica de Catalunya, Campus Nord, Edifici C2, C/ Jordi Girona 1 i 3, E-08034 Barcelona, Spain;Departamento de Matemática Aplicada I, Universidad de Sevilla, Avda Reina Mercedes s/n, E-41012 Sevilla, Spain

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

We denote by ex(n;{C"3,C"4,...,C"s}) or f"s(n) the maximum number of edges in a graph of order n and girth at least s+1. First we give a method to transform an n-vertex graph of girth g into a graph of girth at least g-1 on fewer vertices. For an infinite sequence of values of n and s@?{4,6,10} the obtained graphs are denser than the known constructions of graphs of the same girth s+1. We also give another different construction of dense graphs for an infinite sequence of values of n and s@?{7,11}. These two methods improve the known lower bounds on f"s(n) for s@?{4,6,7,10,11} which were obtained using different algorithms. Finally, to know how good are our results, we have proved that lim sup"n"-"~f"s(n)n^1^+^2^s^-^1=2^-^1^-^2^s^-^1 for s@?{5,7,11}, and s^-^1^-^2^s@?lim sup"n"-"~f"s(n)n^1^+^2^s@?0.5 for s@?{6,10}.