Algorithmic Search for Extremal Graphs of Girth At Least Five (original) (raw)

Electronic Notes in Discrete Mathematics, 2009

By the extremal number ex(n; t) = ex(n; {C 3 , C 4 ,. .. , C t }) we denote the maximum size (number of edges) in a graph of n vertices, n > t, and girth (length of shortest cycle) at least g ≥ t + 1. In 1975, Erdős proposed the problem of determining the extremal numbers ex(n; 4) of a graph of n vertices 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(n; 6) for n = 29, 30 and 31 is equal to 45, 47 and 49, respectively.

On the minimum order of extremal graphs to have a prescribed girth

2007

We show that any n-vertex extremal graph G without cycles of length at most k has girth exactly k +1 if k ≥ 6 and n > (2(k −2) k−2 +k −5)/(k −3). This result provides an improvement of the asymptotical known result by Lazebnik and Wang [J. Graph Theory, 26 (1997), pp. 147-153] who proved thatthe girth is exactly k +1 if k ≥ 12 and n ≥ 2 a 2 +a+1 k a , where a = k − 3 − (k − 2)/4 . Moreover, we prove that the girth of G is at most k + 2 if n > (2(t − 2) k−2 + t − 5)/(t − 3), where t = (k + 1)/2 ≥ 4. In general, for k ≥ 5 we show that the girth of G is at most 2k − 4 if n ≥ 2k − 2.

On the structure of extremal graphs of high girth

Journal of Graph Theory, 1997

Let n ≥ 3 be a positive integer, and let G be a simple graph of order v containing no cycles of length smaller than n + 1 and having the greatest possible number of edges (an extremal graph). Does G contain an n + 1-cycle? In this paper we establish some properties of extremal graphs and present several results where this question is answered affirmatively. For example, this is always the case for (i) v ≥ 8 and n = 5, or (ii) when v is large compared to n: v ≥ 2 a 2 +a+1 n a , where a = n − 3 − n−2 4 , n ≥ 12. On the other hand we prove that the answer to the question is negative for v = 2n + 2 ≥ 26.

On the girth of extremal graphs without shortest cycles

Discrete Mathematics, 2008

Let E X (ν; {C 3 , . . . , C n }) denote the set of graphs G of order ν that contain no cycles of length less than or equal to n which have maximum number of edges. In this paper we consider a problem posed by several authors: does G contain an n + 1 cycle? We prove that the diameter of G is at most n − 1, and present several results concerning the above question: the girth of G is g = n + 1 if (i) ν ≥ n + 5, diameter equal to n − 1 and minimum degree at least 3; (ii) ν ≥ 12, ν ∈ {15, 80, 170} and n = 6. Moreover, if ν = 15 we find an extremal graph of girth 8 obtained from a 3-regular complete bipartite graph subdividing its edges. (iii) We prove that if ν ≥ 2n − 3 and n ≥ 7 the girth is at most 2n − 5. We also show that the answer to the question is negative for ν ≤ n + 1 + (n − 2)/2 .

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

2012

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 +

Girth of -free extremal graphs

Discrete Applied Mathematics, 2012

On size, order, diameter and edge-connectivity of graphs

Acta Mathematica Hungarica, 2017

To bound the size (the number of edges) of a graph in terms of other parameters of a graph forms an important family of problems in the extremal graph theory. We present a number of upper bounds on the size of general graphs and triangle-free graphs. We bound the size of any graph and of any triangle-free graph in terms of its order (number of vertices), diameter and edge-connectivity. We also give an upper bound on the size of triangle-free graphs of given order, diameter and minimum degree. All bounds presented in this paper are asymptotically sharp.

On some variations of extremal graph problems

Discussiones Mathematicae Graph Theory, 1997

A set P of graphs is termed hereditary property if and only if it contains all subgraphs of any graph G belonging to P. A graph is said to be maximal with respect to a hereditary property P (shortly P-maximal) whenever it belongs to P and none of its proper supergraphs of the same order has the property P. A graph is P-extremal if it has a the maximum number of edges among all P-maximal graphs of given order. The number of its edges is denoted by ex(n, P). If the number of edges of a P-maximal graph is minimum, then the graph is called P-saturated and its number of edges is denoted by sat(n, P). In this paper, we consider two famous problems of extremal graph theory. We shall translate them into the language of P-maximal graphs and utilize the properties of the lattice of all hereditary properties in order to establish some general bounds and particular results. Particularly, we shall investigate the behaviour of sat(n, P) and ex(n, P) in some interesting intervals of the mentioned lattice.

Remarks on Extremal Overfull Graphs

An overfull graph is a graph whose number of its edges is greater than the product of its maximum degree and [ ] 2 n , where is the number of vertices. In this paper, some extremals of overfull graphs are presented. We also classify all plannar overfull graphs. n

On some interconnections between combinatorial optimization and extremal graph theory

Yugoslav Journal of Operations Research, 2004

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions.

Algorithmic Search for Extremal Graphs of Girth At Least Five (original) (raw)

Related papers