Math.Comput.Sci. (2010) 3:119–126
DOI 10.1007/s11786-009-0009-6 Mathematics in Computer Science
New Results on EX Graphs
Jianmin Tang · Yuqing Lin · Mirka Miller
Received: 1 March 2009 / Revised: 25 June 2009 / Accepted: 30 July 2009 / Published online: 27 November 2009
© Birkhäuser Verlag Basel/Switzerland 2009
Abstract By the extremal number ex (n; t ) = ex (n;{C
3
, C
4
,..., C
t
}) we denote the maximum size (that is,
number of edges) in a graph of order n > t and girth at least g ≥ t + 1. The set of all the graphs of order n,
containing no cycles of length ≤ t , and of size ex (n; t ), is denoted by EX (n; t ) = EX (n;{C
3
, C
4
,..., C
t
}), these
graphs are called EX graphs. In 1975, Erd˝ os proposed the problem of determining the extremal numbers ex (n; 4)
of a graph of order n and girth at least 5. In this paper, we consider a generalized version of this problem, for t ≥ 5.
In particular, we prove that ex (29; 6) = 45, also we improve some lower bounds and upper bounds of ex
u
(n; t ),
for some particular values of n and t .
Keywords EX graph · Extremal number · Cages
Mathematics Subject Classification (2000) 05C35
1 Introduction
Throughout this paper, only undirected simple graphs without loops or multiple edges are considered. Unless stated
otherwise, we follow [3] for terminology and definitions.
The vertex set (respectively, edge set ) of a graph G is denoted by V (G) (respectively, E (G)). The order (respec-
tively, size) of a graph G is denoted by n = n(G) =|V (G)| (respectively, e = e(G) =| E (G)|). The set of
vertices adjacent to a vertex v is called the neighourhood of v, denoted by N (v). The degree of a vertex v is
deg(v) =| N (v)|. We denote by δ(G) the minimum degree of G and by (G) the maximum degree of G. A graph
G is called k-regular when all the vertices in G have the same degree k . The distance d (u ,v) of two vertices u and
v in V (G) is the length of a shortest path between u and v. The diameter of a graph G, denoted by D is defined
J. Tang (B ) · Y. Lin · M. Miller
School of Electrical Engineering and Computer Science, The University of Newcastle,
Newcastle, NSW 2308, Australia
e-mail: jianmin.tang@newcastle.edu.au
Y. Lin
e-mail: yuqing.lin@newcastle.edu.au
M. Miller
e-mail: mirka.miller@newcastle.edu.au