Preamble

A LOWER BOUND OF THE NUMBER OF EDGES IN A GRAPH CONTAINING NO TWO CYCLES OF THE SAME LENGTH
Chunhui Lai ∗
Dept. of Math., Zhangzhou Teachers College, Zhangzhou, Fujian 363000, P. R. of CHINA.
zjlaichu@public.zzptt.fj.cn
Submitted: November 3, 2000; Accepted: October 20, 2001.

MR Subject Classifications: 05C38, 05C35
Key words: graph, cycle, number of edges

Abstract

In 1975, P. Erdős proposed the problem of determining the maximum number ( f(n) ) of edges in a graph of ( n ) vertices in which any two cycles are of different lengths. In this paper, it is proved that ( f(n) \geq n + 32t - 1 ) for ( t = 27720r + 169 ) (( r \geq 1 )) and ( n \geq 6911 ) (cid:113) (\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{2}} + \frac{2562}{16}t^2 + 514441t - 3309665). Consequently,

Introduction

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, Erdős raised the problem of determining ( f(n) ) (see [1], p.247, Problem 11). Shi [2] proved that ( f(n) \geq n + \left[\left(\sqrt{8n - 23} + 1\right)/2\right] ) for ( n \geq 3 ). Lai [3,4,5,6] proved that for ( n \geq \frac{1381}{9}t^2 + \frac{26}{45}t + \frac{98}{45} ), ( t = 360q + 7 ), ( f(n) \geq n + 19t - 1 ), and for ( n \geq \frac{e^{2m}(2m + 3)}{4} ), ( f(n) < n - 2 + ) (cid:113) ( n\ln\left(\frac{4n}{2m + 3}\right) + 2n + \log_2(n + 6) ). Boros, Caro, Füredi and Yuster [7] proved that ( f(n) \leq n + 1.98\sqrt{n}(1 + o(1)) ).

Let ( v(G) ) denote the number of vertices, and ( \varepsilon(G) ) denote the number of edges. In this paper, we construct a graph ( G ) having no two cycles with the same length which leads to the following result.

Theorem. Let ( t = 27720r + 169 ) (( r \geq 1 )), then for ( n \geq 6911 ) (cid:113) ( \frac{16}{t^2 + 514441t - 3309665} ), ( f(n) \geq n + 32t - 1 ).

2 Proof of Theorem

Proof. Let ( t = 27720r + 169 ), ( r \geq 1 ), ( n_t = 6911 ) (cid:113) ( \frac{16}{t^2 + 514441t - 3309665} ), and ( n \geq n_t ). We shall show that there exists a graph ( G ) on ( n ) vertices with ( n + 32t - 1 ) edges such that all cycles in ( G ) have distinct lengths.

Now we construct the graph ( G ) which consists of a number of subgraphs: ( B_i ) for ( \frac{7t+1}{8} \leq i \leq t - 742 ), ( 58 \leq i \leq \frac{7t-7}{8} ), ( 21t - 1481 \leq i \leq 22t - 798 ), ( 24t - 531 \leq i \leq 24t + 57 ), ( 25t - 741 \leq i \leq 25t + 58 ), ( 26t - 740 \leq i \leq 26t + 57 ), ( 27t - 741 \leq i \leq 27t + 57 ), ( 28t - 741 \leq i \leq 28t + 52 ), ( 29t - 746 \leq i \leq 29t + 60 ), ( 30t - 738 \leq i \leq 30t + 60 ), and ( 31t - 738 \leq i \leq 31t + 799 ).

Now we define these ( B_i )'s. These subgraphs all have a common vertex ( x ); otherwise their vertex sets are pairwise disjoint.

For ( \frac{7t+1}{8} \leq i \leq t - 742 ), let the subgraph ( B_{19t+2i+1} ) consist of a cycle ( C_{19t+2i+1} = xx_1^i \ldots x_{144t+13i+1463} ) and eleven paths sharing a common vertex ( x ), whose other end vertices are on the cycle ( C_{19t+2i+1} ): ( x_{(11t-1)/2}^{i,1}x_2^{i,1}\ldots x_{(13t-1)/2}^{i,2}\ldots x_{(13t-1)/2}^{i,3}\ldots x_{(15t-1)/2}^{i,4}\ldots x_{(15t-1)/2}^{i,5}\ldots x_{(15t-1)/2}^{i,2}x_2^{i,3}x_2^{i,4}x_2^{i,5}x_2^{i,6}x_2^{i,7}x_2^{i,8}x_2^{i,9}x_2^{i,6}\ldots x_{(17t-1)/2}^{i,7}\ldots x_{(17t-1)/2}^{i,8}\ldots x_{(19t-1)/2}^{i,9}\ldots x_{(19t-1)/2}^{i,10}\ldots x_{(21t-1)/2}^{i,11}\ldots x_{(t-571)/2}^{i,10}x_2^{i,11}x_2^{x_{(31t-115)/2+i}}x_{(51t-103)/2+2i}x_{(71t+315)/2+3i}x_{(91t+313)/2+4i}x_{(111t+313)/2+5i}x_{(131t+311)/2+6i}x_{(151t+309)/2+7i}x_{(171t+297)/2+8i}x_{(191t+301)/2+9i}x_{(211t+305)/2+10i}x_{(251t+2357)/2+11i} ).

From the construction, we notice that ( B_{19t+2i+1} ) contains exactly seventy-eight cycles of lengths: ( 21t + i - 57 ), ( 22t + i + 7 ), ( 25t + i + 1 ), ( 26t + i ), ( 29t + i + 3 ), ( 30t + i + 3 ), ( 32t + 2i - 51 ), ( 32t + 2i + 216 ), ( 36t + 2i ), ( 36t + 2i - 1 ), ( 40t + 2i + 5 ), ( 40t + 2i + 744 ), ( 43t + 3i + 215 ), ( 44t + 3i + 209 ), ( 47t + 3i - 7 ), ( 48t + 3i - 4 ), ( 58t + 4i + 1314 ), ( 53t + 4i + 157 ), ( 55t + 4i - 2 ), ( 57t + 4i - 7 ), ( 59t + 4i + 740 ), ( 68t + 5i + 1316 ), ( 65t + 5i + 207 ), ( 66t + 5i - 8 ), ( 69t + 5i + 739 ), ( 77t + 6i + 1310 ), ( 76t + 6i + 201 ), ( 76t + 6i - 6 ), ( 87t + 7i + 1309 ), ( 84t + 7i + 155 ), ( 87t + 7i - 4 ), ( 88t + 7i + 738 ), ( 95t + 8i + 209 ), ( 97t + 8i + 205 ), ( 105t + 9i + 151 ), ( 106t + 9i + 211 ), ( 116t + 10i + 952 ), ( 116t + 10i + 153 ), ( 134t + 12i + 1522 ), ( 144t + 13i + 1464 ), ( 24t + i ), ( 23t + i + 210 ), ( 28t + i - 5 ), ( 27t + i ), ( 19t + 2i + 1 ), ( 31t + i + 742 ), ( 34t + 2i ), ( 34t + 2i + 209 ), ( 38t + 2i - 3 ), ( 38t + 2i - 6 ), ( 42t + 3i + 158 ), ( 49t + 3i + 1312 ), ( 46t + 3i - 1 ), ( 45t + 3i - 1 ), ( 50t + 3i + 746 ), ( 49t + 3i - 1 ), ( 55t + 4i + 208 ), ( 53t + 4i + 215 ), ( 59t + 4i - 2 ), ( 57t + 4i - 5 ), ( 64t + 5i + 214 ), ( 63t + 5i + 157 ), ( 68t + 5i - 3 ), ( 67t + 5i - 5 ), ( 74t + 6i + 213 ), ( 74t + 6i + 156 ), ( 78t + 6i + 738 ), ( 78t + 6i - 3 ), ( 86t + 7i + 203 ), ( 85t + 7i + 207 ), ( 95t + 8i + 149 ), ( 96t + 8i + 1308 ), ( 106t + 9i + 1308 ), ( 97t + 8i + 737 ), ( 107t + 9i + 946 ), ( 115t + 10i + 1307 ), ( 125t + 11i + 1516 ), and ( 126t + 11i + 894 ).

Similarly, for ( 58 \leq i \leq \frac{7t-7}{8} ), let the subgraph ( B_{21t+i-57} ) consist of a cycle ( C_{21t+i-57} = xy_1^i \ldots y_{126t+11i+893} ) and ten paths sharing a common vertex ( x ), whose other end vertices are on the cycle ( C_{21t+i-57} ): ( y_{(11t-1)/2}^{i,1}y_2^{i,1}\ldots y_{(13t-1)/2}^{i,2}\ldots y_{(13t-1)/2}^{i,3}\ldots y_{(15t-1)/2}^{i,4}\ldots y_{(15t-1)/2}^{i,5}\ldots y_{(15t-1)/2}^{i,2}y_2^{i,3}y_2^{i,4}y_2^{i,5}y_2^{i,6}y_2^{i,7}y_2^{i,8}y_2^{i,9}y_2^{i,6}\ldots y_{(17t-1)/2}^{i,7}\ldots y_{(17t-1)/2}^{i,8}\ldots y_{(19t-1)/2}^{i,9}\ldots y_{(19t-1)/2}^{i,10}\ldots y_{(21t-1)/2}^{i,10}y_2^{y_{(31t-115)/2+i}}y_{(51t-103)/2+2i}y_{(71t+315)/2+3i}y_{(91t+313)/2+4i}y_{(111t+313)/2+5i}y_{(131t+311)/2+6i}y_{(151t+309)/2+7i}y_{(171t+297)/2+8i}y_{(191t+301)/2+9i}y_{(211t+305)/2+10i} ).

Based on the construction, ( B_{21t+i-57} ) contains exactly sixty-six cycles of lengths: ( 22t + i + 7 ), ( 21t + i - 57 ), ( 26t + i ), ( 25t + i + 1 ), ( 30t + i + 3 ), ( 29t + i + 3 ), ( 34t + 2i + 209 ), ( 32t + 2i + 216 ), ( 38t + 2i - 6 ), ( 36t + 2i - 1 ), ( 42t + 3i + 158 ), ( 40t + 2i + 744 ), ( 46t + 3i - 1 ), ( 45t + 3i - 1 ), ( 50t + 3i + 746 ), ( 49t + 3i - 1 ), ( 55t + 4i - 2 ), ( 55t + 4i + 208 ), ( 59t + 4i + 740 ), ( 59t + 4i - 2 ), ( 66t + 5i - 8 ), ( 65t + 5i + 207 ), ( 74t + 6i + 156 ), ( 69t + 5i + 739 ), ( 78t + 6i - 3 ), ( 76t + 6i - 6 ), ( 86t + 7i + 203 ), ( 85t + 7i + 207 ), ( 95t + 8i + 209 ), ( 95t + 8i + 149 ), ( 105t + 9i + 151 ), ( 106t + 9i + 211 ), ( 116t + 10i + 952 ), ( 126t + 11i + 894 ), ( 23t + i + 210 ), ( 27t + i ), ( 31t + i + 742 ), ( 34t + 2i ), ( 38t + 2i - 3 ), ( 43t + 3i + 215 ), ( 47t + 3i - 7 ), ( 53t + 4i + 157 ), ( 57t + 4i - 7 ), ( 63t + 5i + 157 ), ( 67t + 5i - 5 ), ( 74t + 6i + 213 ), ( 78t + 6i + 738 ), ( 87t + 7i - 4 ), ( 97t + 8i + 205 ), ( 107t + 9i + 946 ), ( 116t + 10i + 153 ), ( 24t + i ), ( 28t + i - 5 ), ( 32t + 2i - 51 ), ( 36t + 2i ), ( 40t + 2i + 5 ), ( 44t + 3i + 209 ), ( 48t + 3i - 4 ), ( 53t + 4i + 215 ), ( 57t + 4i - 5 ), ( 64t + 5i + 214 ), ( 68t + 5i - 3 ), ( 76t + 6i + 201 ), ( 84t + 7i + 155 ), ( 88t + 7i + 738 ), and ( 97t + 8i + 737 ).

( B_0 ) is a path with an end vertex ( x ) and length ( n - n_t ). Other ( B_i ) are simply cycles of length ( i ).

It is easy to see that
[
v(G) = v(B_0) + \sum_{i=\frac{7t+1}{8}}^{t-742} (v(B_{19t+2i+1}) - 1) + \sum_{i=58}^{\frac{7t-7}{8}} (v(B_{21t+i-57}) - 1) + \sum_{i=21t-1481}^{22t-798} (v(B_i) - 1) + \sum_{i=24t-531}^{24t+57} (v(B_i) - 1) + \sum_{i=25t-741}^{25t+58} (v(B_i) - 1) + \sum_{i=26t-740}^{26t+57} (v(B_i) - 1) + \sum_{i=27t-741}^{27t+57} (v(B_i) - 1) + \sum_{i=28t-741}^{28t+52} (v(B_i) - 1) + \sum_{i=29t-746}^{29t+60} (v(B_i) - 1) + \sum_{i=30t-738}^{30t+60} (v(B_i) - 1) + \sum_{i=31t-738}^{31t+799} (v(B_i) - 1)
]
[
= n - n_t + 1 + \sum_{i=\frac{7t+1}{8}}^{t-742} (144t + 13i + 1463 - 1) + \sum_{i=58}^{\frac{7t-7}{8}} (126t + 11i + 893 - 1) + \sum_{i=21t-1481}^{22t-798} (i - 1) + \sum_{i=24t-531}^{24t+57} (i - 1) + \sum_{i=25t-741}^{25t+58} (i - 1) + \sum_{i=26t-740}^{26t+57} (i - 1) + \sum_{i=27t-741}^{27t+57} (i - 1) + \sum_{i=28t-741}^{28t+52} (i - 1) + \sum_{i=29t-746}^{29t+60} (i - 1) + \sum_{i=30t-738}^{30t+60} (i - 1) + \sum_{i=31t-738}^{31t+799} (i - 1)
]
[
= n - n_t + \frac{16(-3309665 + 1028882t + 6911t^2)}{16} = n.
]

Now we compute the number of edges of ( G ):
[
\varepsilon(G) = \varepsilon(B_0) + \sum_{i=\frac{7t+1}{8}}^{t-742} \varepsilon(B_{19t+2i+1}) + \sum_{i=58}^{\frac{7t-7}{8}} \varepsilon(B_{21t+i-57}) + \sum_{i=21t-1481}^{22t-798} \varepsilon(B_i) + \sum_{i=24t-531}^{24t+57} \varepsilon(B_i) + \sum_{i=25t-741}^{25t+58} \varepsilon(B_i) + \sum_{i=26t-740}^{26t+57} \varepsilon(B_i) + \sum_{i=27t-741}^{27t+57} \varepsilon(B_i) + \sum_{i=28t-741}^{28t+52} \varepsilon(B_i) + \sum_{i=29t-746}^{29t+60} \varepsilon(B_i) + \sum_{i=30t-738}^{30t+60} \varepsilon(B_i) + \sum_{i=31t-738}^{31t+799} \varepsilon(B_i)
]
[
= n - n_t + \sum_{i=\frac{7t+1}{8}}^{t-742} (144t + 13i + 1464) + \sum_{i=58}^{\frac{7t-7}{8}} (126t + 11i + 894) + \sum_{i=21t-1481}^{22t-798} i + \sum_{i=24t-531}^{24t+57} i + \sum_{i=25t-741}^{25t+58} i + \sum_{i=26t-740}^{26t+57} i + \sum_{i=27t-741}^{27t+57} i + \sum_{i=28t-741}^{28t+52} i + \sum_{i=29t-746}^{29t+60} i + \sum_{i=30t-738}^{30t+60} i + \sum_{i=31t-738}^{31t+799} i
]
[
= n - n_t + \frac{16(-3309681 + 1029394t + 6911t^2)}{16} = n + 32t - 1.
]

Then ( f(n) \geq n + 32t - 1 ) for ( n \geq n_t ). This completes the proof of the theorem.

From the above theorem, we have
[
\liminf_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq \text{(cid:115)} \text{(cid:113)} \frac{2 + 2562}{16},
]
which is better than the previous bounds ( \sqrt{2} ) (see [2]), and ( \sqrt{2 + \frac{487}{1381}} ) (see [6]).

Combining this with Boros, Caro, Füredi and Yuster's upper bound, we have
[
1.98 \geq \limsup_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq \liminf_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq 1.5397.
]

Acknowledgment

The author thanks Prof. Yair Caro and Raphael Yuster for sending reference [7]. The author also thanks Prof. Cheng Zhao for his advice.

References

[1] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976).
[2] Y. Shi, On maximum cycle-distributed graphs, Discrete Math. 71(1988) 57-71.
[3] Chunhui Lai, On the Erdős problem, J. Zhangzhou Teachers College (Natural Science Edition) 3(1)(1989) 55-59.
[4] Chunhui Lai, Upper bound and lower bound of ( f(n) ), J. Zhangzhou Teachers College (Natural Science Edition) 4(1)(1990) 29,30-34.
[5] Chunhui Lai, On the size of graphs with all cycle having distinct length, Discrete Math. 122(1993) 363-364.
[6] Chunhui Lai, The edges in a graph in which no two cycles have the same length, J. Zhangzhou Teachers College (Natural Science Edition) 8(4)(1994), 30-34.
[7] E. Boros, Y. Caro, Z. Füredi and R. Yuster, Covering non-uniform hypergraphs (submitted, 2000).