Girth of {C3,...,Cs} -free extremal graphs

Authors:
E. Abajo;C. Balbuena;A. Diánez
Affiliations:
Departamento de Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes 2, 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 2, E-41012 Sevilla, Spain
Venue:
Discrete Applied Mathematics
Year:
2012

Citing 8
Cited 0

Extremal graphs without three-cycles or four-cycles

Journal of Graph Theory
On the structure of extremal graphs of high girth

Journal of Graph Theory
Graphs and Digraphs, Fourth Edition

Graphs and Digraphs, Fourth Edition
On the Minimum Order of Extremal Graphs to have a Prescribed Girth

SIAM Journal on Discrete Mathematics
Calculating the extremal number ex(v;{C3,C4,...,Cn})

Discrete Applied Mathematics
On Moore graphs with diameters 2 and 3

IBM Journal of Research and Development
New families of graphs without short cycles and large size

Discrete Applied Mathematics
Exact values of ex(ν;{C3,C4,...,Cn})

Discrete Applied Mathematics

Quantified Score

Hi-index	0.04

Visualization

Abstract

Let G be a {C"3,...,C"s}-free graph with as many edges as possible. In this paper we consider a question studied by several authors, the compulsory existence of the cycle C"s"+"1 in G. The answer has already been proved to be affirmative for s=3,4,5,6. In this work we show that the girth of G is g(G)=s+1 when the order of G is at least 1+s(s-22)^s^-^2-4s-4 if s is even, and 1+(s-1)^3((s-2)^2-14)^s^-^3^2-8s2(s-2)^2-10 if s is odd. This bound is an improvement of the best general result so far known. Moreover, we also prove in the case s=7 that the girth is g(G)=8 for order at least 14 and characterize all the extremal graphs whose girth is not 8.