Preamble

On the Number of Edges in Some Graphs
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. The sequence $(c_1, c_2, \cdots, c_n)$ represents the cycle length distribution of a graph $G$ with $n$ vertices, where $c_i$ is the number of cycles of length $i$ in $G$. Let $f(a_1, a_2, \cdots, a_n)$ denote the maximum possible number of edges in a graph satisfying $c_i \leq a_i$, where $a_i$ is a nonnegative integer. In 1991, Shi posed the problem of determining $f(a_1, a_2, \cdots, an)$, which extended Erdős's problem. It is clear that $f(n) = f(1, 1, \cdots, 1)$. Let $g(n, m) = f(a_1, a_2, \cdots, a_n)$, where $a_i = 1$ if $i/m$ is an integer, and $a_i = 0$ otherwise. It is clear that $f(n) = g(n, 1)$.

We prove that
$$\liminf_{n\to\infty} \frac{g(n,m)-n}{\sqrt{n}} \geq \sqrt{2 + \frac{7654}{99}} \approx 2.444$$
for all even integers $m$, which improves upon previous bounds (Lai, 2017). We show that
$$\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} \geq \sqrt{2 + \frac{7654}{19071}}$$
which is better than the previous bounds: $\sqrt{2}$ (Shi, 1988) and earlier results. We make the following conjecture:
$$\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} = \sqrt{2 + \frac{7654}{19071}}.$$

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

Funding: Project supported by the National Science Foundation of China (No. 61379021; No. 11401290), NSF of Fujian (2015J01018; 2018J01423), Fujian Provincial Training Foundation for "Bai-Quan-Wan Talents Engineering", Project of Fujian Education Department (JZ160455), the Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics, Project supported by Minnan Normal University.

Email address: laich2011@msn.cn; laichunhui@mnnu.edu.cn (Chunhui Lai)

Introduction

Let $f(n)$ be the maximum number of edges in a graph with $n$ vertices in which no two cycles have the same length. In 1975, Erdős raised the problem of determining $f(n)$ (see Bondy and Murty \cite{BondyMurty1976}, p. 247, Problem 11). Shi \cite{Shi1988} proved the following lower bound:

Theorem 1 (Shi \cite{Shi1988}). $f(n) \geq n + \left\lfloor\left(\sqrt{8n-23}+1\right)/2\right\rfloor$ for $n \geq 3$.

Additional related results were obtained by Chen, Lehel, Jacobson and Shreve \cite{ChenLehelJacobsonShreve1998}, Jia \cite{Jia1996}, Lai \cite{Lai1993, Lai2001, Lai2003, Lai2017}, and Shi \cite{Shi1992, Shi1994}. Boros, Caro, Füredi and Yuster \cite{BorosCaroFurediYuster2001} proved the following upper bound:

Theorem 2 (Boros, Caro, Füredi and Yuster \cite{BorosCaroFurediYuster2001}). For $n$ sufficiently large, $f(n) < n + 1.98\sqrt{n}$.

Lai \cite{Lai2017} improved Shi's lower bound as follows:

Theorem 3 (Lai \cite{Lai2017}). Let $t = 1260r + 169$ ($r \geq 1$), then
$$f(n) \geq n + \frac{t^2 + 87978t + 15957}{4t + 1}$$
for $n \geq \frac{2119}{4}t^2 + \frac{26399}{2}t + 6932215$.

Lai \cite{Lai1993} proposed the following conjecture:

Conjecture 4 (Lai \cite{Lai1993}).
$$\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} = \sqrt{2}.$$

It would be nice to prove that $\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} > \sqrt{2}$. Survey papers on this problem can be found in Tian \cite{Tian1986}, Zhang \cite{Zhang2007}, and Lai and Liu \cite{LaiLiu2014}. The progress on all 50 problems in \cite{BondyMurty1976} can be found in Locke \cite{Locke}.

The sequence $(c_1, c_2, \cdots, c_n)$ is the cycle length distribution of a graph $G$ with $n$ vertices, where $c_i$ is the number of cycles of length $i$ in $G$. Let $f(a_1, a_2, \cdots, a_n)$ denote the maximum possible number of edges in a graph which satisfies $c_i \leq a_i$, where $a_i$ is a nonnegative integer. Shi \cite{Shi1991} posed the problem of determining $f(a_1, a_2, \cdots, a_n)$, which extended the problem due to Erdős. It is clear that $f(n) = f(1, 1, \cdots, 1)$. Let $g(n, m) = f(a_1, a_2, \cdots, a_n)$, where $a_i = 1$ if $i/m$ is an integer, and $a_i = 0$ otherwise. It is clear that $f(n) = g(n, 1)$.

In this paper, we obtain the following results:

Theorem 5. Let $m$ be even, $s_1 > s_2$, $s_1 + 3s_2 > k$, then
$$g(n, m) \geq n + (k + s_1 + 2s_2 + 1)t - 1$$
for $n \geq \left(\frac{3}{4}mk^2 + \frac{1}{2}mks_1 + \frac{3}{4}mk + \frac{1}{2}mks_2 + \frac{1}{2}ms_1s_2 + \frac{9}{4}ms_2 - k - s_1 - 2s_2 + \frac{1}{2}m - 1\right)t + 1$.

Theorem 6. Let $t = 1260r + 169$ ($r \geq 1$), then for $n \geq \frac{1309}{2}t^2 - \frac{1349159}{3}t + 6932215$,
$$f(n) \geq n + \frac{119}{3}t - 26399.$$

Proof of Theorem 5

Proof. Let
$$n_t = \left(\frac{3}{4}mk^2 + \frac{1}{2}mks_1 + \frac{3}{4}mk + \frac{1}{2}mks_2 + \frac{1}{2}ms_1s_2 + \frac{9}{4}ms_2 - k - s_1 - 2s_2 + \frac{1}{2}m - 1\right)t + 1,$$
where $m$ is even, $s_1 > s_2$, $s_1 + 3s_2 > k$, and $n \geq n_t$. It suffices to show that there exists a graph $G$ on $n$ vertices with $n + (k + s_1 + 2s_2 + 1)t - 1$ edges such that all cycles in $G$ have distinct lengths and all cycle lengths are multiples of $m$.

We construct the graph $G$ consisting of several subgraphs $B_i$ for $0 \leq i \leq s_1t$, $i = s_1t + j$ ($1 \leq j \leq s_2t$), and $i = s_1t + s_2t + j$ ($1 \leq j \leq t$). These subgraphs all share a single common vertex $x$, and are otherwise vertex-disjoint.

For $1 \leq i \leq s_2t$, let the subgraph $B_{s_1t+i}$ consist of a cycle
$$C_{s_1t+i} = xa_1^i a_2^i \cdots a_{ms_1t+2ms_2t+mi-1}^i x$$
and a path
$$P_{s_1t+i} = a_{ms_1t+ms_2t+mi}^i \cdots a_{ms_1t+2ms_2t+mi-2}^i.$$
Based on this construction, $B_{s_1t+i}$ contains exactly three cycles of lengths $ms_1t + mi$, $ms_1t + ms_2t + mi$, and $ms_1t + 2ms_2t + mi$.

For $1 \leq i \leq t$, let the subgraph $B_{s_1t+s_2t+i}$ consist of a cycle
$$C_{s_1t+s_2t+i} = xy_1^i \cdots y_{ms_1t+3ms_2t+mk(k+1)t+mi-1}^i x$$
and $k$ paths sharing the common vertex $x$, with their other endpoints on the cycle $C_{s_1t+s_2t+i}$:
$$P_{s_1t+s_2t+i,p} = y_{ms_1t+3ms_2t-mkt+m(p-1)t+mi}^i \cdots y_{ms_1t+3ms_2t+mk(2p-1)t+m(p-1)t+mi}^i \quad (p = 1, 2, \ldots, k).$$

As a cycle with $k$ chords contains exactly $\binom{k+2}{2}$ distinct cycles, $B_{s_1t+s_2t+i}$ contains cycles of lengths $ms_1t + 3ms_2t + mkht + (h + j - 1)mt + mi$ for $j \geq 1$, $h \geq 0$, and $k + 1 \geq j + h$.

$B_0$ is a path with one endpoint at $x$ and length $n - n_t$. All other $B_i$ are simply cycles of length $mi$.

Therefore, $g(n, m) \geq n + (k + s_1 + 2s_2 + 1)t - 1$ for $n \geq n_t$. This completes the proof.

From Theorem 5, we have
$$\liminf_{n\to\infty} \frac{g(n,m)-n}{\sqrt{n}} \geq \sqrt{\frac{k + s_1 + 2s_2 + 1}{\frac{3}{4}mk^2 + \frac{1}{2}mks_1 + \frac{3}{4}mk + \frac{1}{2}mks_2 + \frac{1}{2}ms_1s_2 + \frac{9}{4}ms_2 + \frac{1}{2}m}}$$
for all even integers $m$.

Let $s_1 = 28499066$, $s_2 = 4749839$, $k = 14249542$. Then
$$\liminf_{n\to\infty} \frac{g(n,m)-n}{\sqrt{n}} \geq \sqrt{2.444}$$
for all even integers $m$.

Proof of Theorem 6

Proof. Let $t = 1260r + 169$ with $r \geq 1$, and let
$$n_t = \frac{1309}{2}t^2 - \frac{1349159}{3}t + 6932215.$$
For $n \geq n_t$, it suffices to show that there exists a graph $G$ on $n$ vertices with $n + \frac{119}{3}t - 26399$ edges such that all cycles in $G$ have distinct lengths.

We construct the graph $G$ consisting of several subgraphs $B_i$ for $0 \leq i \leq 22t$, $i = 22t + j$ ($1 \leq j \leq \frac{5t-8}{3}$), $i = 23t + \frac{2t-2}{3} + j$ ($1 \leq j \leq \frac{5t-8}{3}$), and $i = 32t + j - 60$ ($58 \leq j \leq t - 742$). These subgraphs all share a single common vertex $x$, and are otherwise vertex-disjoint.

For $1 \leq i \leq \frac{5t-8}{3}$, let the subgraph $B_{22t+i}$ consist of a cycle
$$C_{22t+i} = xa_1^i a_2^i \cdots a_{28t+\frac{2t-2}{3}+2i-3}^i x$$
and a path
$$P_{22t+i} = a_{56t-2}^i \cdots a_{76t-4}^i.$$
Based on this construction, $B_{22t+i}$ contains exactly three cycles of lengths $22t + i$, $25t + \frac{t-1}{3} + i - 1$, and $28t + \frac{2t-2}{3} + 2i - 2$.

For $1 \leq i \leq \frac{5t-8}{3}$, let the subgraph $B_{23t+\frac{2t-2}{3}+i}$ consist of a cycle
$$C_{23t+\frac{2t-2}{3}+i} = xb_1^i b_2^i \cdots b_{28t+\frac{2t-2}{3}+2i-2}^i x$$
and a path
$$P_{23t+\frac{2t-2}{3}+i} = b_{11t-1}^i \cdots b_{76t-4}^i.$$
Based on this construction, $B_{23t+\frac{2t-2}{3}+i}$ contains exactly three cycles of lengths $23t + \frac{2t-2}{3} + i$, $27t + i - 1$, and $28t + \frac{2t-2}{3} + 2i - 1$.

For $58 \leq i \leq t - 742$, let the subgraph $B_{32t+i-60}$ consist of a cycle
$$C_{32t+i-60} = xy_1^i \cdots y_{137t+11i+890}^i x$$
and ten paths sharing the common vertex $x$, with their other endpoints on the cycle $C_{32t+i-60}$:
\begin{align}
& P_{32t+i-60,1} = y_{11t-2}^i \cdots y_{21t-59+i}^i, \
& P_{32t+i-60,2} = y_{12t-2}^i \cdots y_{31t-53+2i}^i, \
& P_{32t+i-60,3} = y_{12t-2}^i \cdots y_{41t+156+3i}^i, \
& P_{32t+i-60,4} = y_{13t-2}^i \cdots y_{51t+155+4i}^i, \
& P_{32t+i-60,5} = y_{13t-2}^i \cdots y_{61t+155+5i}^i, \
& P_{32t+i-60,6} = y_{14t-2}^i \cdots y_{71t+154+6i}^i, \
& P_{32t+i-60,7} = y_{14t-2}^i \cdots y_{81t+153+7i}^i, \
& P_{32t+i-60,8} = y_{15t-2}^i \cdots y_{91t+147+8i}^i, \
& P_{32t+i-60,9} = y_{15t-2}^i \cdots y_{101t+149+9i}^i, \
& P_{32t+i-60,10} = y_{16t-2}^i \cdots y_{111t+151+10i}^i.
\end{align}

As a cycle with $d$ chords contains exactly $\binom{d+2}{2}$ distinct cycles, $B_{32t+i-60}$ contains cycles of lengths: $32t + i - 60$, $33t + i + 4$, $34t + i + 207$, $35t + i - 3$, $36t + i - 2$, $37t + i - 3$, $38t + i - 3$, $39t + i - 8$, $40t + i$, $41t + i$, $42t + i + 739$, $43t + 2i - 54$, $43t + 2i + 213$, $45t + 2i + 206$, $45t + 2i - 3$, $47t + 2i - 3$, $47t + 2i - 4$, $49t + 2i - 9$, $49t + 2i - 6$, $51t + 2i + 2$, $51t + 2i + 741$, $53t + 3i + 155$, $54t + 3i + 212$, $55t + 3i + 206$, $56t + 3i - 4$, $57t + 3i - 4$, $58t + 3i - 10$, $59t + 3i - 7$, $60t + 3i - 4$, $61t + 3i + 743$, $64t + 4i + 154$, $64t + 4i + 212$, $66t + 4i + 205$, $66t + 4i - 5$, $68t + 4i - 10$, $68t + 4i - 8$, $70t + 4i - 5$, $70t + 4i + 737$, $74t + 5i + 154$, $75t + 5i + 211$, $76t + 5i + 204$, $77t + 5i - 11$, $78t + 5i - 8$, $79t + 5i - 6$, $80t + 5i + 736$, $85t + 6i + 153$, $85t + 6i + 210$, $87t + 6i + 198$, $87t + 6i - 9$, $89t + 6i - 6$, $89t + 6i + 735$, $95t + 7i + 152$, $96t + 7i + 204$, $97t + 7i + 200$, $98t + 7i - 7$, $99t + 7i + 735$, $106t + 8i + 146$, $106t + 8i + 206$, $108t + 8i + 202$, $108t + 8i + 734$, $116t + 9i + 148$, $117t + 9i + 208$, $118t + 9i + 943$, $127t + 10i + 150$, $127t + 10i + 949$, $137t + 11i + 891$.

$B_0$ is a path with one endpoint at $x$ and length $n - n_t$. All other $B_i$ are simply cycles of length $i$.

Therefore, $f(n) \geq n + \frac{119}{3}t - 26399$ for $n \geq n_t$. This completes the proof.

From Theorem 6, we have
$$\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} \geq \sqrt{2 + \frac{7654}{19071}},$$
which is better than the previous bounds: $\sqrt{2}$ (see \cite{Shi1988}) and $\sqrt{2 + \frac{7654}{19071}}$ (see \cite{Lai2017}). Combining this with Boros, Caro, Füredi and Yuster's upper bound (Theorem 2), we obtain
$$1.98 \geq \limsup_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} \geq \liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} \geq \sqrt{2 + \frac{7654}{19071}}.$$

From the proof of Theorem 6, we have
$$\liminf_{n\to\infty} \frac{g(n,m)-n}{\sqrt{n}} \geq \sqrt{2 + \frac{7654}{19071}}$$
for all integers $m$.

If $m = 1$ and $1 \leq i \leq t$, we could construct a subgraph similar to $B_{s_1t+s_2t+i}$ consisting of a cycle $C_{s_1t+s_2t+i}$ and $k$ paths sharing a common vertex $x$, with their other endpoints on the cycle $C_{s_1t+s_2t+i}$, such that all cycles in $B_{s_1t+s_2t+i}$ have distinct lengths. This would yield $\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} \geq \sqrt{2.444} > \sqrt{2 + \frac{7654}{19071}}$. However, we have only constructed such a subgraph for $m = 1$ and $58 \leq i \leq t - 742$, using a cycle with ten paths sharing a common vertex $x$ to obtain $\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} \geq \sqrt{2 + \frac{7654}{19071}}$.

Since the liminf for $\frac{g(n,m)-n}{\sqrt{n}}$ for even $m$ is $\sqrt{2.444}$, it is reasonable to suspect that such a lower bound also holds for $\frac{f(n)-n}{\sqrt{n}}$. We make the following conjecture:

Conjecture 7.
$$\liminf_{n\to\infty} \frac{f(n)-n}{\sqrt{n}} = \sqrt{2.444}.$$

Acknowledgment

The author would like to thank Professor Endre Boros, Yair Caro, Ronald J. Gould, Gyula O.H. Katona, and Raphael Yuster for their advice. The author would also like to thank the referees for their many valuable comments and suggestions.

References

[1] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976). MR3138588

[2] E. Boros, Y. Caro, Z. Füredi and R. Yuster, Covering non-uniform hypergraphs, J. Combin. Theory Ser. B 82(2001), 270-284. MR1842115 (2002g:05130)

[3] G. Chen, J. Lehel, M. S. Jacobson and W. E. Shreve, Note on graphs without repeated cycle lengths, J. Graph Theory 29(1998), 11-15. MR1633908 (2000d:05059)

[4] X. Jia, Some extremal problems on cycle distributed graphs, Congr. Numer. 121(1996), 216-222. MR1431994 (97i:05068)

[5] C. Lai, On the size of graphs with all cycle having distinct length, Discrete Math. 122(1993) 363-364. MR1246693(94i:05048)

[6] C. Lai, A lower bound for the number of edges in a graph containing no two cycles of the same length, Electron. J. Combin. 8(2001), Note 9, 1-6. MR1877662 (2002k:05124)

[7] C. Lai, Graphs without repeated cycle lengths, Australas. J. Combin. 27(2003), 101-105. MR1955391 (2003m:05102)

[8] C. Lai, On the size of graphs without repeated cycle lengths, Discrete Appl. Math. 232 (2017), 226-229. MR3711962

[9] C. Lai, M. Liu, Some open problems on cycles, J. Combin. Math. Combin. Comput. 91(2014), 51-64. MR3287706

[10] S. C. Locke, Unsolved problems: http://math.fau.edu/locke/Unsolved.htm

[11] Y. Shi, On maximum cycle-distributed graphs, Discrete Math. 71(1988), 57-71. MR0954686 (89i:05169)

[12] Y. Shi, Some problems of cycle length distribution, J. Nanjing Univ. (Natural Sciences), Special Issue On Graph Theory, 27(1991), 233-234.

[13] Y. Shi, The number of edges in a maximum cycle distributed graph, Discrete Math. 104(1992), 205-209. MR1172850 (93d:05083)

[14] Y. Shi, On simple MCD graphs containing a subgraph homeomorphic to $K_4$, Discrete Math. 126(1994), 325-338. MR1264498 (95f:05072)

[15] F. Tian, The progress on some problems in graph theory, Qufu Shifan Daxue Xuebao Ziran Kexue Ban 1986, no. 2, 30-36. MR0865617

[16] K. Zhang, Progress of some problems in graph theory, J. Math. Res. Exposition 27(3) (2007), 563-576. MR2349503