Preamble

Chunhui Lai
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, Fujian, P.R. China

Abstract

In 1975, P. Erdős proposed the problem of determining the maximum number $f(n)$ of edges in a graph with $n$ vertices in which any two cycles have different lengths. In this paper, we prove that for $t = 1260r + 169$ ($r \geq 1$) and $n \geq 2119$, $f(n) \geq n + \frac{4t^2 + 87978t + 15957}{19071}$. Consequently, $\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} \geq \sqrt{\frac{4}{19071}}$, which improves upon the previous bounds of $2$ (see Shi, Discrete Math. 71(1988), 57-71) and $2 + \frac{2}{5}$ (see Lai, Australas. J. Combin. 27(2003), 101-105). This result disproves Conjecture 4 of Lai.

Key words: Graph, cycle, number of edges.
AMS 2000 MSC: 05C38, 05C35.

Introduction

Let $f(n)$ denote the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. In 1975, Erdős posed the problem of determining $f(n)$ (see Bondy and Murty [1], p.247, Problem 11). Shi [15] established the following lower bound:

Theorem 1 (Shi [15]). $f(n) \geq n + \left\lfloor\left(\sqrt{8n - 23} + 1\right)/2\right\rfloor$ for $n \geq 3$.

Additional related results were obtained by Chen, Lehel, Jacobson, and Shreve [3], Jia [5], Lai [6–8], and Shi [16,18–20]. Boros, Caro, Füredi, and Yuster [2] proved an upper bound:

Theorem 2 (Boros, Caro, Füredi and Yuster [2]). For $n$ sufficiently large, $f(n) < n + 1.98\sqrt{n}$.

Lai [9] improved the lower bound:

Theorem 3 (Lai [9]). $f(n) \geq n + \sqrt{2n(1 - o(1))}$.

Lai [6,9] proposed the following conjectures:

Conjecture 4 (Lai [9]). $\liminf_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} = 2$.

Conjecture 5 (Lai [6]). $\liminf_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq 2 + \frac{2}{5}$.

Markström [13] raised the following problem:

Problem 6 (Markström [13]). Determine the maximum number of edges in a Hamiltonian graph on $n$ vertices with no repeated cycle lengths.

Results for the maximum number of edges in a 2-connected graph on $n$ vertices with distinct cycle lengths can be found in [2,3,15]. Survey articles on this problem appear in Tian [21], Zhang [24], and Lai and Liu [10]. The progress on all 50 problems from [1] can be found in Locke [12].

A related topic concerns the Entringer problem, which asks which simple graphs have exactly one cycle of each length $\ell$ for $3 \leq \ell \leq v$ (see Problem 10 in [1]). This problem was posed in 1973 by R.C. Entringer. For developments on this topic, see [4,11,13,14,17,22,23].

In this paper, we construct a graph $G$ with no two cycles of the same length, yielding the following result:

Theorem 7. Let $t = 1260r + 169$ ($r \geq 1$). Then $f(n) \geq n + \frac{4t^2 + 87978t + 15957}{19071}$ for $n \geq 2119$. Consequently, Conjecture 4 is false.

Proof of Theorem 7

Proof. Let $t = 1260r + 169$ with $r \geq 1$, and let $n_t = 2119$. We shall show that for $n \geq n_t$, there exists a graph $G$ on $n$ vertices with $n + \frac{4t^2 + 87978t + 15957}{19071}$ edges such that all cycles in $G$ have distinct lengths.

We construct the graph $G$ from several subgraphs $B_i$ defined for indices $i$ belonging to the following sets:
- $0 \leq i \leq 20t$
- $27t \leq i \leq 28t + 64$
- $29t - 734 \leq i \leq 29t + 267$
- $30t - 531 \leq i \leq 30t + 57$
- $31t - 741 \leq i \leq 31t + 58$
- $32t - 740 \leq i \leq 32t + 57$
- $33t - 741 \leq i \leq 33t + 57$
- $34t - 741 \leq i \leq 34t + 52$
- $35t - 746 \leq i \leq 35t + 60$
- $36t - 738 \leq i \leq 36t + 60$
- $37t - 738 \leq i \leq 37t + 799$
- $i = 20t + j$ for $1 \leq j \leq \frac{t-7}{6}$
- $i = 20t + \frac{t-1}{6}$
- $i = 20t + \frac{t-1}{3}$
- $i = 20t + \frac{t-1}{2}$
- $i = 20t + \frac{2t-2}{3}$
- $i = 21t - 2$
- $i = 21t - 1$
- $i = 21t + 2j + 1$ for $0 \leq j \leq t - 1$
- $i = 21t + 2j$ for $0 \leq j \leq \frac{t-3}{2}$
- $i = 23t + 2j + 1$ for $0 \leq j \leq \frac{t-3}{2}$
- $i = 26t$

All these subgraphs share a single common vertex $x$, and their vertex sets are otherwise pairwise disjoint.

Construction of the subgraphs:

For $1 \leq i \leq \frac{t-7}{6}$, let $B_{20t+i}$ consist of a cycle $C_{20t+i} = xa_1^i a_2^i \cdots a_{\frac{62t-8}{3}+2i}^i$ and a path $a_1^i a_2^i \cdots a_{\frac{59t-5}{3}}^i a_{\frac{61t-1}{3}}^i$. By construction, $B_{20t+i}$ contains exactly three cycles of lengths $20t + i$, $20t + \frac{t-1}{6} + i - 1$, and $20t + \frac{2t-2}{3} + 2i - 1$.

For $1 \leq i \leq \frac{t-7}{6}$, let $B_{20t+\frac{t-1}{6}+i}$ consist of a cycle $C_{20t+\frac{t-1}{6}+i} = xb_1^i b_2^i \cdots b_{\frac{62t-5}{3}+2i}^i$ and a path $b_1^i b_2^i \cdots b_{10t-1}^i b_{\frac{61t-1}{3}}^i$. This subgraph contains exactly three cycles of lengths $20t + \frac{t-1}{6} + i$, $20t + \frac{t-1}{3} + i$, and $20t + \frac{2t-2}{3} + 2i$.

For $0 \leq i \leq t - 1$, let $B_{21t+2i+1}$ consist of a cycle $C_{21t+2i+1} = xu_1^i u_2^i \cdots u_{25t+2i-1}^i$ and a path $u_1^i u_2^i \cdots u_{\frac{19t+2i-1}{2}}^i u_{\frac{23t+2i+1}{2}}^i$. This subgraph contains exactly three cycles of lengths $21t + 2i + 1$, $23t + 2i$, and $25t + 2i$.

For $0 \leq i \leq \frac{t-3}{2}$, let $B_{21t+2i}$ consist of a cycle $C_{21t+2i} = xv_1^i v_2^i \cdots v_{25t+2i}^i$ and a path $v_1^i v_2^i \cdots v_{9t+i-1}^i v_{12t+i}^i$. This subgraph contains exactly three cycles of lengths $21t + 2i$, $22t + 2i + 1$, and $25t + 2i + 1$.

For $i = \frac{t-1}{2}$, $B_{21t+2i}$ is simply a cycle of length $22t - 1$.

For $0 \leq i \leq \frac{t-3}{2}$, let $B_{23t+2i+1}$ consist of a cycle $C_{23t+2i+1} = xw_1^i w_2^i \cdots w_{26t+2i+1}^i$ and a path $w_1^i w_2^i \cdots w_{\frac{21t+2i-1}{2}}^i w_{\frac{25t+2i+1}{2}}^i$. This subgraph contains exactly three cycles of lengths $23t + 2i + 1$, $24t + 2i + 2$, and $26t + 2i + 2$.

For $i = \frac{t-1}{2}$, $B_{23t+2i+1}$ is simply a cycle of length $24t$.

For $58 \leq i \leq t - 742$, let $B_{27t+i-57}$ consist of a cycle $C_{27t+i-57} = xy_1^i y_2^i \cdots y_{132t+11i+893}^i$ and ten paths sharing the common vertex $x$, with their other endpoints on the cycle $C_{27t+i-57}$:
- $y_1^i y_2^i \cdots y_{\frac{17t-1}{2}}^i$
- $y_2^i y_3^i \cdots y_{\frac{19t-1}{2}}^i$
- $y_3^i y_4^i \cdots y_{\frac{19t-1}{2}}^i$
- $y_4^i y_5^i \cdots y_{\frac{21t-1}{2}}^i$
- $y_5^i y_6^i \cdots y_{\frac{21t-1}{2}}^i$
- $y_6^i y_7^i \cdots y_{\frac{23t-1}{2}}^i$
- $y_7^i y_8^i \cdots y_{\frac{23t-1}{2}}^i$
- $y_8^i y_9^i \cdots y_{\frac{25t-1}{2}}^i$
- $y_9^i y_{10}^i \cdots y_{\frac{25t-1}{2}}^i$
- $y_{10}^i y_{11}^i \cdots y_{\frac{27t-1}{2}}^i y_{\frac{37t-115}{2}+i} y_{\frac{57t-103}{2}+2i} y_{\frac{77t+315}{2}+3i} y_{\frac{97t+313}{2}+4i} y_{\frac{117t+313}{2}+5i} y_{\frac{137t+311}{2}+6i} y_{\frac{157t+309}{2}+7i} y_{\frac{177t+297}{2}+8i} y_{\frac{197t+301}{2}+9i} y_{\frac{217t+305}{2}+10i}$

A cycle with $d$ chords contains exactly $\binom{d+2}{2}$ distinct cycles. Therefore, $B_{27t+i-57}$ contains cycles of the following 66 lengths: $27t + i - 57$, $28t + i + 7$, $29t + i + 210$, $30t + i$, $31t + i + 1$, $32t + i$, $33t + i$, $34t + i - 5$, $35t + i + 3$, $36t + i + 3$, $37t + i + 742$, $38t + 2i - 51$, $38t + 2i + 216$, $40t + 2i + 209$, $40t + 2i$, $42t + 2i$, $42t + 2i - 1$, $44t + 2i - 6$, $44t + 2i - 3$, $46t + 2i + 5$, $46t + 2i + 744$, $48t + 3i + 158$, $49t + 3i + 215$, $50t + 3i + 209$, $51t + 3i - 1$, $52t + 3i - 1$, $53t + 3i - 7$, $54t + 3i - 4$, $55t + 3i - 1$, $56t + 3i + 746$, $59t + 4i + 157$, $59t + 4i + 215$, $61t + 4i + 208$, $61t + 4i - 2$, $63t + 4i - 7$, $63t + 4i - 5$, $65t + 4i - 2$, $65t + 4i + 740$, $69t + 5i + 157$, $70t + 5i + 214$, $71t + 5i + 207$, $72t + 5i - 8$, $73t + 5i - 5$, $74t + 5i - 3$, $75t + 5i + 739$, $80t + 6i + 156$, $80t + 6i + 213$, $82t + 6i + 201$, $82t + 6i - 6$, $84t + 6i - 3$, $84t + 6i + 738$, $90t + 7i + 155$, $91t + 7i + 207$, $92t + 7i + 203$, $93t + 7i - 4$, $94t + 7i + 738$, $101t + 8i + 149$, $101t + 8i + 209$, $103t + 8i + 205$, $103t + 8i + 737$, $111t + 9i + 151$, $112t + 9i + 211$, $113t + 9i + 946$, $122t + 10i + 153$, $122t + 10i + 952$, and $132t + 11i + 894$.

Finally, $B_0$ is a path with one endpoint at $x$ and length $n - n_t$, while each remaining $B_i$ is simply a cycle of length $i$.

This construction yields $f(n) \geq n + \frac{107}{3}$ for $n \geq n_t$, completing the proof.

From Theorem 7, we obtain $\liminf_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq \sqrt{\frac{4}{19071}}$, which improves upon the previous bounds of $2$ (see [15]) and $2 + \frac{2}{5}$ (see [9]). Therefore, Conjecture 4 is false. Combining this with the upper bound of Boros, Caro, Füredi, and Yuster (Theorem 2), we have:
$$1.98 \geq \limsup_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq \liminf_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq \sqrt{\frac{4}{19071}}.$$

Acknowledgment

The author would like to thank the referees for their many valuable comments and suggestions.

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