Skip to main content
SpringerLink
Log in

On the Existence of Regular Sparse Anti-magic Squares of Odd Order

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers n and d with \(d<n\), an \(n\times n\) array A based on \(\{0,1,\ldots ,nd\}\) is called a sparse anti-magic square of order n with density d, denoted by SAMS(n, d), if each element of \(\{1,2,\ldots ,nd\}\) occurs exactly one entry of A, and its row-sums, column-sums and two main diagonal-sums constitute a set of \(2n+2\) consecutive integers. An SAMS(n, d) is called regular if there are exactly d non-zero elements (or positive entries) in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of odd order, and it is proved that for any odd n, there exists a regular SAMS(n, d) if and only if \(2\le d\le n-1\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abe, G.: Unsolved problems on magic squares. Discret. Math. 127, 3–13 (1994)

    Article  MathSciNet  Google Scholar 

  2. Ahmed, M.: Algebraic combinatorics of magic squares. Ph.D Dissertation, University of California, Davis (2004)

  3. Andrews, W.S.: Magic Squares and Cubes, 2nd edn. Dover, New York (1960)

    MATH  Google Scholar 

  4. Chen, G., Chen, H., Chen, K., Li, W.: Regular sparse anti-magic squares with small odd densities. Discret. Math. 339, 138–156 (2016)

    Article  MathSciNet  Google Scholar 

  5. Chen, G., Li, W., Zhong, M., Xin, B.: Constructions of regular sparse anti-magic squares. arXiv:2002.07357

  6. Chen, K., Chen, G., Li, W.: Regular sparse anti-magic squares with the second maximum density. Linear Algebra Appl. 457, 12–28 (2014)

    Article  MathSciNet  Google Scholar 

  7. Chen, K., Li, W., Chen, G., Wei, R.: Regular sparse anti-magic squares with maximum density. Ars Comb. 126, 167–183 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Cormie, J., Linek, V., Jiang, S., Chen, R.: Investigating the antimagic square. J. Comb. Math. Comb. Comput. 43, 175–197 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Gray, I.D.: Vertex-magic total labelings of regular graphs. SIAM J. Discret. Math. 21(1), 170–177 (2007)

    Article  MathSciNet  Google Scholar 

  10. Gray, I.D., MacDougall, J.A.: Sparse anti-magic squares and vertex-magic labelings of bipartite graphs. Discret. Math. 306, 2878–2892 (2006)

    Article  MathSciNet  Google Scholar 

  11. Gray, I.D., MacDougall, J.A.: Sparse semi-magic squares and vertex-magic labelings. Ars Comb. 80, 225–242 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Gray, I.D., MacDougall, J.A.: Vertex-magic labeling of regular graphs: disjoint unions and assemblages. Discret. Appl. Math. 160, 1114–1125 (2012)

    Article  MathSciNet  Google Scholar 

  13. Gray, I.D., MacDougall, J.A.: Vertex-magic labelings of regular graphs. II. Discret. Math. 309(20), 5986–5999 (2009)

    Article  MathSciNet  Google Scholar 

  14. Gray, I.D., MacDougall, J.A., Simpson, R.J., Wallis, W.D.: Vertex-magic total labelings of complete bipartite graphs. Ars Comb. 69, 117–127 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Jiang, S.: Anti-magic squares of even order. J. Kodai Math. 25, 1–7 (2002)

    Article  MathSciNet  Google Scholar 

  16. Kotzig, A.: On Magic Valuations of Trichromatic Graphs. Reports of the CRM, CRM-148, December (1971)

  17. Kudrle, J.M., Menard, S.B.: Magic square. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn., pp. 524–528. CRC Press, Boca Raton (2006)

    Google Scholar 

  18. Li, W., Chen, K., Su, R.: Existence of regular sparse magic squares. Acta Math. Appl. Sin. (Chin. Ser.) 34, 1118–1135 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Liang, Y., Wong, T., Zhu, X.: Anti-magic labeling of trees. Discret. Math. 331, 9–14 (2014)

    Article  MathSciNet  Google Scholar 

  20. Liang, Y., Zhu, X.: Anti-magic labeling of cubic graphs. J. Graph Theory 75, 31–36 (2014)

    Article  MathSciNet  Google Scholar 

  21. Wallis, W.D.: Magic Graphs. Birkhäuser, Basel (2001)

    Book  Google Scholar 

  22. Wallis, W.D.: Vertex magic labelings of multiple graphs. Congr. Numer. 152, 81–83 (2001)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Professor Zhu Lie of Suzhou University for his encouragement and many helpful suggestions. The authors thank the anonymous reviewers for their careful check, constructive comments and suggestions that greatly improved the quality of this paper.

Funding

This work was supported by National Natural Science Foundation of China (Grant nos. 11871417, 11501181 and 71962030). This work was also supported by Youth Fund of Humanities and Social Sciences, Ministry of Education (Grant no. 20YJC910009).

Author information

Authors and Affiliations

  1. Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, Henan, People’s Republic of China

    Guangzhou Chen

  2. School of Information Science and Technology, Xichang University, Xichang, 615000, Sichuan, People’s Republic of China

    Wen Li

  3. Central Primary School of Tingzi Town, Dazhou, 635011, Sichuan, People’s Republic of China

    Ming Zhong

  4. School of Sciences, Xichang University, Xichang, 615000, Sichuan, People’s Republic of China

    Bangying Xin

Authors
  1. Guangzhou Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wen Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ming Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bangying Xin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangzhou Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix 1

\((n,d)=(21,16)\)

Note that the last column is the row-sums, the last row is the column-sums, the left diagonal-sum is 2696 and the right diagonal-sum is 2718.

Appendix 2

Example 2

There exists a regular SAMS(n, d) for \((n,d)\in \{(9,6),(15,5),(27,22)\}\).

Proof

For \((n,d)=(9,6)\), we write \(d=t+2\), then \(t=d-2=4\), by Lemma 4.3, there exists an SFD(4, 9) over [1, 36] below.

$$\begin{aligned} C=\left( \begin{array}{ccccccccc} 1 &{} 2 &{} 3 &{} 4 &{} 6 &{} 7 &{} 8 &{} 9 &{} 5 \\ 5 &{} 9 &{} 8 &{} 7 &{} 6 &{} 4 &{} 3 &{} 2 &{} 1 \\ 9 &{} 8 &{} 7 &{} 6 &{} 4 &{} 3 &{} 2 &{} 1 &{} 5 \\ 5 &{} 1 &{} 2 &{} 3 &{} 4 &{} 6 &{} 7 &{} 8 &{} 9 \\ \end{array}\right) +9\left( \begin{array}{ccccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 2 &{} 2 &{} 2 &{} 2 &{} 2 &{} 2 &{} 2 \\ 3 &{} 3 &{} 3 &{} 3 &{} 3 &{} 3 &{} 3 \\ \end{array}\right) =\left( \begin{array}{ccccccccc} 1 &{} 2 &{} 3 &{} 4 &{} 6 &{} 7 &{} 8 &{} 9 &{} 5 \\ 14 &{} 18 &{} 17 &{} 16 &{} 15 &{} 13 &{} 12 &{} 11 &{} 10 \\ 27 &{} 26 &{} 25 &{} 24 &{} 22 &{} 21 &{} 20 &{} 19 &{} 23 \\ 32 &{} 28 &{} 29 &{} 30 &{} 31 &{} 33 &{} 34 &{} 35 &{} 36 \\ \end{array}\right) . \end{aligned}$$

Let \(C^*=(c_{i,j}^*)\), where \( c^*_{i,j}= \left\{ \begin{array}{ll} c_{i,j}, &{}\quad \textit{if} \ i=1,2, \\ c_{i,j}+2n, &{}\quad \textit{if} \ i=3,4.\\ \end{array} \right. \)

$$\begin{aligned} C^*= \left( \begin{array}{ccccccccc} 1 &{} 2 &{} 3 &{} 4 &{} 6 &{} 7 &{} 8 &{} 9 &{} 5 \\ 14 &{} 18 &{} 17 &{} 16 &{} 15 &{} 13 &{} 12 &{} 11 &{} 10 \\ 45 &{} 44 &{} 43 &{} 42 &{} 40 &{} 39 &{} 38 &{} 37 &{} 41 \\ 50 &{} 46 &{} 47 &{} 48 &{} 49 &{} 51 &{} 52 &{} 53 &{} 54 \\ \end{array} \right) . \end{aligned}$$

We get E below by putting the element \(c_{i,j}^*\) into the cell of B as the way of the proof of Theorem 4.11.

The arrays A and W are the same as Theorem 2.2, and the array U is listed below by the proof of Theorem 4.11.

$$\begin{aligned} A=\left( \begin{array}{ccccccccc} 10 &{}11&{}12&{} 14 &{} 15 &{} 16&{}17 &{} 18 &{} 9 \\ 1 &{} 2 &{} 3 &{} 4 &{} 13 &{} 5 &{} 6 &{} 7 &{} 8 \\ \end{array} \right) , \end{aligned}$$

Then

is a regular SAMS(9, 6).

For \((n,d)=(15,5)\), let \(W^*\) be the regular SAMS(15, 2) from Theorem 2.2, which is the same as Example 1. Let B be the Latin square of order 15 and \(D=(d_{i,j})\) be the regular SMS(15, 3) by Lemma 4.9. Construct \(D^*=(d_{i,j}^*)\), where \(d_{i.j}^*=d_{i,j}+30\), and \(D^*\) is listed below.

Here, above empty cells all indicate 0. It is easy to check that \(D^*+W^*\) is a regular SAMS(15, 5).

For \((n,d)=(27,22)\). We shall obtain an SAMS(27, 22) by dividing two steps in the following.

Step 1: Construct a special SAMS(27, 10).

By Remark 3, there exists an SFD(8, 27), \(C=(c_{i,j})\), which is listed below.

$$\begin{aligned} C=\left( \begin{array}{ccccccccccccccccccccccccccc} 1&{} 2&{} 3&{} 4&{} 5&{} 6&{} 7&{} 8&{} 9&{} 10&{} 11&{} 12&{} 13&{} 15&{} 16&{} 17&{} 18&{} 19&{} 20&{} 21&{} 22&{} 23&{} 24&{} 25&{} 26&{} 27&{} 14\\ 41&{} 54&{} 53&{} 52&{} 51&{} 50&{} 49&{} 48&{} 47&{} 46&{} 45&{} 44&{} 43&{} 42&{} 40&{} 39&{} 38&{} 37&{} 36&{} 35&{} 34&{} 33&{} 32&{} 31&{} 30&{} 29&{} 28\\ 81&{} 80&{} 79&{} 78&{} 77&{} 76&{} 75&{} 74&{} 73&{} 72&{} 71&{} 70&{} 69&{} 67&{} 66&{} 65&{} 64&{} 63&{} 62&{} 61&{} 60&{} 59&{} 58&{} 57&{} 56&{} 55&{} 68\\ 95&{} 82&{} 83&{} 84&{} 85&{} 86&{} 87&{} 88&{} 89&{} 90&{} 91&{} 92&{} 93&{} 94&{} 96&{} 97&{} 98&{} 99&{} 100&{}101&{}102&{}103&{}104&{}105&{}106&{}107&{}108\\ 109&{}110&{}111&{}112 &{}113&{}114&{}115&{}116&{}117&{}118&{}119&{}120&{}121&{}123&{}124&{}125&{}126&{}127&{}128&{}129&{}130&{}131&{}132&{}133&{}134&{}135&{}122\\ 149&{}162&{}161&{}160&{}159&{}158&{}157&{}156&{}155&{}154&{}153&{}152&{}151&{}150&{}148&{}147&{}146&{}145&{}144&{}143&{}142&{}141&{}140&{}139&{}138&{}137&{}136\\ 189&{}188&{}187&{}186&{}185&{}184&{}183&{}182&{}181&{}180&{}179&{}178&{}177&{}175&{}174&{}173&{}172&{}171&{}170&{}169&{}168&{}167&{}166&{}165&{}164&{}163&{}176\\ 203&{}190&{}191&{}192&{}193&{}194&{}195&{}196&{}197&{}198&{}199&{}200&{}201&{}202&{}204&{}205&{}206&{}207&{}208&{}209&{}210&{}211&{}212&{}213&{}214&{}215&{}216\\ \end{array} \right) . \end{aligned}$$

By the proof of Theorem 4.14, the array \(C^*=(c^*_{i,j})\) is obtained as follows, where

$$\begin{aligned} c^*_{i,j}= \left\{ \begin{array}{ll} c_{i,j}, &{}\quad \text{if } i\in \{1,2,3,4\} \\ c_{i,j}+54, &{}\quad \text{if } i\in \{5,6,7,8\}. \end{array} \right. \end{aligned}$$
$$\begin{aligned} C^*=\left( \begin{array}{ccccccccccccccccccccccccccc} 1&{} 2&{} 3&{} 4&{} 5&{} 6&{} 7&{} 8&{} 9&{} 10&{} 11&{} 12&{} 13&{} 15&{} 16&{} 17&{} 18&{} 19&{} 20&{} 21&{} 22&{} 23&{} 24&{} 25&{} 26&{} 27&{} 14\\ 41&{} 54&{} 53&{} 52&{} 51&{} 50&{} 49&{} 48&{} 47&{} 46&{} 45&{} 44&{} 43&{} 42&{} 40&{} 39&{} 38&{} 37&{} 36&{} 35&{} 34&{} 33&{} 32&{} 31&{} 30&{} 29&{} 28\\ 81&{} 80&{} 79&{} 78&{} 77&{} 76&{} 75&{} 74&{} 73&{} 72&{} 71&{} 70&{} 69&{} 67&{} 66&{} 65&{} 64&{} 63&{} 62&{} 61&{} 60&{} 59&{} 58&{} 57&{} 56&{} 55&{} 68\\ 95&{} 82&{} 83&{} 84&{} 85&{} 86&{} 87&{} 88&{} 89&{} 90&{} 91&{} 92&{} 93&{} 94&{} 96&{} 97&{} 98&{} 99&{} 100&{}101&{}102&{}103&{}104&{}105&{}106&{}107&{}108\\ 163&{} 164&{} 165&{} 166&{} 167&{} 168&{} 169&{} 170&{} 171&{} 172&{} 173&{} 174&{} 175&{} 177&{} 178&{} 179&{} 180&{} 181&{} 182&{} 183&{} 184&{} 185&{} 186&{} 187&{} 188&{} 189&{} 176\\ 203&{} 216&{} 215&{} 214&{} 213&{} 212&{} 211&{} 210&{} 209&{} 208&{} 207&{} 206&{} 205&{} 204&{} 202&{} 201&{} 200&{} 199&{} 198&{} 197&{} 196&{} 195&{} 194&{} 193&{} 192&{} 191&{} 190\\ 243&{} 242&{} 241&{} 240&{} 239&{} 238&{} 237&{} 236&{} 235&{} 234&{} 233&{} 232&{} 231&{} 229&{} 228&{} 227&{} 226&{} 225&{} 224&{} 223&{} 222&{} 221&{} 220&{} 219&{} 218&{} 217&{} 230\\ 257&{} 244&{} 245&{} 246&{} 247&{} 248&{} 249&{} 250&{} 251&{} 252&{} 253&{} 254&{} 255&{} 256&{} 258&{} 259&{} 260&{} 261&{} 262&{} 263&{} 264&{} 265&{} 266&{} 267&{} 268&{} 269&{} 270\\ \end{array} \right) . \end{aligned}$$

Then \(E'\) and \(W^*\) are obtained by the proof of Theorem 4.14.

Then \(Q=E'+W^*\) is a regular SAMS(27, 10).

Step 2: Construct the array D in the proof of Theorem 4.14. The arrays \(C_1,C_2\) and \(C_3\) are an SFD(3, 27) over [271, 351], an SFD(6, 27) over [352, 513] and an SFD(3, 27) over [514, 594] respectively, which are listed in the following.

$$\begin{aligned} C_1=\left( \begin{array}{ccccccccccccccccccccccccccc} 297&{} 283&{} 296&{} 282&{} 295&{} 281&{} 294&{} 280&{} 293&{} 279&{} 292&{} 278&{} 291&{} 277&{} 290&{} 276&{} 289&{} 275&{} 288&{} 274&{} 287&{} 273&{} 286&{} 272&{} 285&{} 271&{} 284\\ 298&{} 299&{} 300&{} 301&{} 302&{} 303&{} 304&{} 305&{} 306&{} 307&{} 308&{} 309&{} 310&{} 311&{} 312&{} 313&{} 314&{} 315&{} 316&{} 317&{} 318&{} 319&{} 320&{} 321&{} 322&{} 323&{} 324\\ 338&{} 351&{} 337&{} 350&{} 336&{} 349&{} 335&{} 348&{} 334&{} 347&{} 333&{} 346&{} 332&{} 345&{} 331&{} 344&{} 330&{} 343&{} 329&{} 342&{} 328&{} 341&{} 327&{} 340&{} 326&{} 339&{} 325\\ \end{array} \right) . \end{aligned}$$
$$\begin{aligned} C_2=\left( \begin{array}{ccccccccccccccccccccccccccc} 378&{} 364&{} 377&{} 363&{} 376&{} 362&{} 375&{} 361&{} 374&{} 360&{} 373&{} 359&{} 372&{} 358&{} 371&{} 357&{} 370&{} 356&{} 369&{} 355&{} 368&{} 354&{} 367&{} 353&{} 366&{} 352&{} 365\\ 379&{} 380&{} 381&{} 382&{} 383&{} 384&{} 385&{} 386&{} 387&{} 388&{} 389&{} 390&{} 391&{} 392&{} 393&{} 394&{} 395&{} 396&{} 397&{} 398&{} 399&{} 400&{} 401&{} 402&{} 403&{} 404&{} 405\\ 419&{} 432&{} 418&{} 431&{} 417&{} 430&{} 416&{} 429&{} 415&{} 428&{} 414&{} 427&{} 413&{} 426&{} 412&{} 425&{} 411&{} 424&{} 410&{} 423&{} 409&{} 422&{} 408&{} 421&{} 407&{} 420&{} 406\\ 459&{} 445&{} 458&{} 444&{} 457&{} 443&{} 456&{} 442&{} 455&{} 441&{} 454&{} 440&{} 453&{} 439&{} 452&{} 438&{} 451&{} 437&{} 450&{} 436&{} 449&{} 435&{} 448&{} 434&{} 447&{} 433&{} 446\\ 460&{} 461&{} 462&{} 463&{} 464&{} 465&{} 466&{} 467&{} 468&{} 469&{} 470&{} 471&{} 472&{} 473&{} 474&{} 475&{} 476&{} 477&{} 478&{} 479&{} 480&{} 481&{} 482&{} 483&{} 484&{} 485&{} 486\\ 500&{} 513&{} 499&{} 512&{} 498&{} 511&{} 497&{} 510&{} 496&{} 509&{} 495&{} 508&{} 494&{} 507&{} 493&{} 506&{} 492&{} 505&{} 491&{} 504&{} 490&{} 503&{} 489&{} 502&{} 488&{} 501&{} 487\\ \end{array} \right) . \end{aligned}$$
$$\begin{aligned} C_3=\left( \begin{array}{ccccccccccccccccccccccccccc} 540&{} 526&{} 539&{} 525&{} 538&{} 524&{} 537&{} 523&{} 536&{} 522&{} 535&{} 521&{} 534&{} 520&{} 533&{} 519&{} 532&{} 518&{} 531&{} 517&{} 530&{} 516&{} 529&{} 515&{} 528&{} 514&{} 527\\ 541&{} 542&{} 543&{} 544&{} 545&{} 546&{} 547&{} 548&{} 549&{} 550&{} 551&{} 552&{} 553&{} 554&{} 555&{} 556&{} 557&{} 558&{} 559&{} 560&{} 561&{} 562&{} 563&{} 564&{} 565&{} 566&{} 567\\ 581&{} 594&{} 580&{} 593&{} 579&{} 592&{} 578&{} 591&{} 577&{} 590&{} 576&{} 589&{} 575&{} 588&{} 574&{} 587&{} 573&{} 586&{} 572&{} 585&{} 571&{} 584&{} 570&{} 583&{} 569&{} 582&{} 568\\ \end{array} \right) . \end{aligned}$$

By Lemma 4.12, there exists a regular SMS(27, 6), denoted by \(D_2\), consisting of the red numbers in the following array, and the array \(D_1\) consist of the yellow and blue numbers in the following array. It is clear that the element sets of \(D_1\) and \(D_2\) are \([271,351]\cup [514,594]\) and [352, 513] respectively, and \(D_1\) and \(D_2\) are compatible. Then one can check that \(D=D_1+D_2\) is a regular SAMS(27, 12) over [271, 594], which is compatible with \(W^*\), so we can obtain a regular SAMS(27, 22) below.

figure a

\(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, G., Li, W., Zhong, M. et al. On the Existence of Regular Sparse Anti-magic Squares of Odd Order. Graphs and Combinatorics 38, 47 (2022). https://doi.org/10.1007/s00373-021-02437-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-021-02437-z

Keywords

Search

Navigation