On the size of graphs with all cycle having distinct length

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Discrete Mathematics 122 (1993) 363-364 North-Holland

363

Note

On the size of graphs with all cycle having distinct length

Chunhui Lai

Department of Mathematics, Zhangzhou Teachers Coliege, Zhangzhou, Fujian 363000, China

Received 18 June 1990 Revised 16 August 1991

Abstract

Let f(n) be the maximum number of edges in a graph on n vertices in which no two cycles have the same length. In 1975, Erdiis raised the question of determiningf(n)(see Bondy and Murty (1976)). In this note, we prove that for n>36.5tZ-4.5t+l one has f(n)>n+9r-1. We conjecture that lim(f(n) -

n,lJ;; < $.

Let f(n) be the maximum number of edges in a graph on n vertices in which no two cycles have the same length. In 1975, Erdiis raised the question of determining f(n) (see 1, p. 247, Problem 111). In 1988, Shi [2] proved that

f(n)3n+[(@ZG+1)/2] for all n33.

He conjectured that

f(n)=n+[($GZG+l)/2] for all n33,

and he proved that this conjecture for 3 dn,< 17 is true. In 1989 and 1990, I [3,4]

proved that

n+(4/15),/%-C<f(n)<n-2+ n. ln(4n/(2m + 3)) + 2n + log,(n + 6) for all n 3 e’“(2m + 3)/4 (m = 1,2,3, . . .), and the conjecture of Yongbing Shi is not true.

In this note n, t, i are all integers.

Correspondence to: Chunhui Lai, Department of Mathematics, Zhangzhou Teachers College, ihangzhou, Fujian 363000, China.

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364 c. Lrri

Theorem. Let t31, then f‘(n)>n+9t-1 ,fbr n336.5t2-4,5t+l.

Proof. We give an example G consisting of 2-connected blocks, Bi, (1 < i < 7~). These blocks all have a common vertex x, otherwise their vertex sets are pairwise disjoint.

For id 6t Bi is simply the cycle of length i. The block Bet + i (for 1~ i < t) is obtained from a cycle

C 13f+3i=XXl,i-Y2.i “. X13t+3ik l,ix

such that the vertex x is connected to Xqt+i,i by a path

XX13t+3i,iX13t+3i+1,i”‘X15t+3i-2,iX4r+i,ir

and by another path

XX15z+3i-l,iX15f+3i,i~“X17r+3i-3,iX71+2i,i

to the vertex X71+zi,i. Here B6t+i contains cycles of lengths 6t + i, 7t + i, 8t + i, 9t + 2i, 1 It +2i, and 13t + 3i. Then the theorem easily follows from the inequality f(n+ l)>.f(n)+ 1. 0

From the theorem of this paper we have lim(f(n)-n)/&>/J2+16/73.

We may make the following conjecture.

Conjecture. lim( f(n)- n)/& < J3.

References

111 PI c-11 141

J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976).

Y. Shi, On maximum cycle-distributed graphs, Discrete Math. 71 (1988) 57-71.

C. Lai, On the ErdGs problem, J. Zhangzhou Teachers College (Natural Science Edition) 1 (1989) 55-59.

C. Lai, Upper bound and lower bound of f’(n), J. Zhangzhou Teachers College (Natural Science Edition) 4 (1990) 29, 30-34.