Preamble

GRAPHS WITHOUT REPEATED CYCLE LENGTHS

Chunhui Lai
Department of Mathematics, Zhangzhou Teachers College, Zhangzhou, Fujian 363000, P. R. China; and Graph Theory and Combinatorics Laboratory, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, P. R. China
zjlaichu@public.zzptt.fj.cn

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

Australasian Journal of Combinatorics 27(2003), 101-105.

Abstract

In 1975, P. Erdős proposed the problem of determining the maximum number $f(n)$ of edges in a graph on $n$ vertices in which any two cycles have different lengths. In this paper, we prove that $f(n) \geq n + 36t$ for $t = 1260r + 169$ ($r \geq 1$) and $n \geq 540t^2 + 175811t + 7989$, which improves upon previous bounds. Consequently, we obtain new results on the asymptotic behavior of $f(n)$ that surpass earlier bounds established in \cite{2} and \cite{7}.

Combining this with the upper bound of Boros, Caro, Füredi and Yuster \cite{8}, we establish that
$$1.98 \geq \limsup_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq \liminf_{n\to\infty} \frac{f(n) - n}{\sqrt{n}}.$$

We propose the following conjecture:

Conjecture. $f(n) - n = \Theta(\sqrt{n})$.

Introduction

Let $f(n)$ denote the maximum number of edges in a graph on $n$ vertices containing no two cycles of the same length. In 1975, Erdős posed the problem of determining $f(n)$ (see \cite{1}, p.247, Problem 11). Shi \cite{2} proved that $f(n) \geq n + \left\lfloor\left(\sqrt{8n - 23} + 1\right)/2\right\rfloor$ for $n \geq 3$. Lai \cite{3,4,5,6,7} established several bounds, showing that for $n \geq \frac{6911}{16}t^2 + 514441t - 3309665$ with $t = 27720r + 169$, and for $n \geq e^{2m}(2m + 3)/4$, we have $f(n) \geq n + 32t - 1$ and $f(n) < n - 2 + \sqrt{n\ln(4n/(2m + 3)) + 2n} + \log_2(n + 6)$. Boros, Caro, Füredi and Yuster \cite{8} proved that $f(n) \leq n + 1.98\sqrt{n}(1 + o(1))$.

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

Theorem. Let $t = 1260r + 169$ ($r \geq 1$). Then $f(n) \geq n + 36t$ for $n \geq 540t^2 + 175811t + 7989$.

2 Proof of the Theorem

Proof. Let $t = 1260r + 169$ with $r \geq 1$, and let $n_t = 540t^2 + 175811t + 7989$. For any $n \geq n_t$, we will construct a graph $G$ on $n$ vertices with exactly $n + 36t$ edges such that all cycles in $G$ have distinct lengths.

The graph $G$ consists of a collection of subgraphs $B_i$ defined for specific index sets. All subgraphs share a common vertex $x$, while their remaining vertex sets are pairwise disjoint. The index set for the subgraphs is:
\begin{align}
&{i \mid 0 \leq i \leq 21t - 1} \cup {i \mid 27t \leq i \leq 28t + 64} \
&\cup {i \mid 29t - 734 \leq i \leq 29t + 267} \cup {i \mid 30t - 531 \leq i \leq 30t + 57} \
&\cup {i \mid 31t - 741 \leq i \leq 31t + 58} \cup {i \mid 32t - 740 \leq i \leq 32t + 57} \
&\cup {i \mid 33t - 741 \leq i \leq 33t + 57} \cup {i \mid 34t - 741 \leq i \leq 34t + 52} \
&\cup {i \mid 35t - 746 \leq i \leq 35t + 60} \cup {i \mid 36t - 738 \leq i \leq 36t + 60} \
&\cup {i \mid 37t - 738 \leq i \leq 37t + 799} \cup {i = 21t + 2j + 1 \mid 0 \leq j \leq t - 1} \
&\cup {i = 21t + 2j \mid 0 \leq j \leq \frac{t-3}{2}} \cup {i = 23t + 2j + 1 \mid 0 \leq j \leq \frac{t-3}{2}} \cup {i = 26t}.
\end{align}

We now define these subgraphs $B_i$ in detail.

For $0 \leq i \leq t - 1$, the subgraph $B_{21t+2i+1}$ consists of a cycle and a path attached to it. Specifically, it contains a cycle $C_{21t+2i+1}$ and a path $P_{21t+2i+1}$ that share some vertices. This construction yields exactly three cycles of lengths $21t + 2i + 1$, $23t + 2i$, and $25t + 2i$.

For $0 \leq i \leq \frac{t-3}{2}$, the subgraph $B_{21t+2i}$ consists of a cycle $C_{21t+2i}$ and a path $P_{21t+2i}$. This subgraph contains exactly three cycles of lengths $21t + 2i$, $22t + 2i + 1$, and $25t + 2i + 1$.

For $0 \leq i \leq \frac{t-3}{2}$, the subgraph $B_{23t+2i+1}$ consists of a cycle $C_{23t+2i+1}$ and a path $P_{23t+2i+1}$. This subgraph contains exactly three cycles of lengths $23t + 2i + 1$, $24t + 2i + 2$, and $26t + 2i + 2$.

For $58 \leq i \leq t - 742$, the subgraph $B_{27t+i-57}$ is more complex. It consists of a cycle $C_{27t+i-57} = xy_1^i \ldots y_{132t+11i+893}^i$ and ten paths that all share the common vertex $x$, with their other endpoints on the cycle $C_{27t+i-57}$. The paths are:
\begin{align}
&y_{1,1}^i y_{2,1}^i \ldots y_{(17t-1)/2}^i, \
&y_{1,2}^i \ldots y_{(19t-1)/2}^i, \
&y_{1,3}^i \ldots y_{(19t-1)/2}^i, \
&y_{1,4}^i \ldots y_{(21t-1)/2}^i, \
&y_{1,5}^i \ldots y_{(21t-1)/2}^i, \
&y_{1,6}^i \ldots y_{(23t-1)/2}^i, \
&y_{1,7}^i \ldots y_{(23t-1)/2}^i, \
&y_{1,8}^i y_{2,8}^i \ldots y_{(25t-1)/2}^i, \
&y_{1,9}^i \ldots y_{(25t-1)/2}^i, \
&y_{1,10}^i \ldots y_{(27t-1)/2}^i.
\end{align}

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

Finally, $B_0$ is defined as a path with one endpoint at $x$ and length $n - n_t$. For all other indices $i$ in the specified ranges, $B_i$ is simply a cycle of length $i$.

By construction, each subgraph $B_i$ contributes cycles of distinct lengths, and cycles belonging to different subgraphs have different lengths because their length expressions involve different linear combinations of $t$ and $i$. Therefore, the graph $G$ contains no two cycles of the same length and has exactly $n + 36t$ edges.

Thus, for all $n \geq n_t$, we have $f(n) \geq n + 36t$, completing the proof.

From the theorem, we obtain
$$\liminf_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq 1.98,$$
which improves upon previous bounds established in \cite{2} and \cite{7}.

Combining this with the upper bound of Boros, Caro, Füredi and Yuster \cite{8}, we have
$$1.98 \geq \limsup_{n\to\infty} \frac{f(n) - n}{\sqrt{n}} \geq \liminf_{n\to\infty} \frac{f(n) - n}{\sqrt{n}}.$$

We propose the following conjecture:

Conjecture. $f(n) - n = \Theta(\sqrt{n})$.

Acknowledgment

The author thanks Prof. Yair Caro and Raphael Yuster for providing reference \cite{8}. The author also thanks Prof. Genghua Fan and Cheng Zhao for their valuable suggestions.

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 number of 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] Chunhui Lai, A lower bound for the number of edges in a graph containing no two cycles of the same length, The Electronic J. of Combinatorics 8(2001), #N9.

[8] E. Boros, Y. Caro, Z. Füredi and R. Yuster, Covering non-uniform hypergraphs (submitted, 2000). Now published in Journal of Combinatorial Theory, Series B 82(2001), 270-284.