Prof Ian M. Wanless

## A matchbox autobiography

I was born and grew up in Canberra, and studied mathematics at the Australian National University. My PhD thesis was entitled "Permanents, matchings and Latin rectangles". It was supervised by Brendan McKay, who to this day makes scurrilous aspersions on my parentage. In 1998-9 I worked with Nick Wormald at the University of Melbourne on asymptotic enumerations and random graphs. Then in October 1999, I took up a Junior Research Fellowship at Christ Church, Oxford which ended in 2003. Then, for old times' sake I spent a year or two back at ANU in Brendan's tender loving care. In 2005 I had a senior lectureship at Charles Darwin University, which turned out to be a bad idea because half way through the year they scrapped all their interesting maths courses. Since 2006 I have been in the School of Mathematical Sciences at Monash University in Melbourne. In 2011 I was lucky enough to be awarded an Australian Research Council "Future Fellowship", which I guess means that I am now a future former future fellow! We have a very active combinatorics group at Monash. Click here to see what we've been up to.

## Research Interests

My research is in combinatorics, which seems to be the study of how many different ways you can define the word `combinatorics'. So let me be more specific. I'm interested mainly in Latin squares, matrix permanents and graph theory.

If you would like to do a project in one of these areas, check out some suggested topics.

## Latin Squares

A Latin square is a matrix in which each number occurs exactly once in every row and once in every column. For example here's a Latin square of order 8:

```          1 5 3 7 2 4 6 8
4 7 5 1 3 6 8 2
3 1 2 4 6 8 5 7
5 3 7 6 8 2 4 1
2 4 6 8 5 1 7 3
7 6 8 2 4 3 1 5
6 8 1 3 7 5 2 4
8 2 4 5 1 7 3 6
```

Notice that each row and column consists of a permutation of the set {1,2,3,...,8}. This particular square has some other nice patterns in it. For example the even numbers occur in cyclic order in each row, whereas the odd numbers occur in cyclic order in each column. This square is also unusual in that it has no Latin subsquares. Here's a research challenge for you: Generalise this pattern to larger Latin squares. I'm particularly interested in orders that are powers of two, but I'm genuinely interested in finding this square's bigger siblings. E-mail me if you have any ideas.

Here's another challenge for you. Brendan and I recently counted the Latin squares of order 11, and it turns out there are rather a lot of them. 776966836171770144107444346734230682311065600000 to be precise!

If we look at the number of reduced latin squares (meaning their first row and column are in natural order) then the numbers are as follows:

```    Order  Number of reduced LS
2     1
3     1
4     22
5     23.7
6     26.3.72
7     210.3.5.1103
8     217.3.1361291
9     221.32.5231.3824477
10    228.32.5.31.37.547135293937
11    235.34.5.2801.2206499.62368028479
```
Question: Why are these numbers divisible by such a high power of two? From theoretical arguments we can only predict that the last number here is divisible by 24, which is a long way short of 235!

From time to time I enumerate classes of latin squares so I have a lot of data sitting on my hard drive. Some of it is available here, but if there's something else you want then ask -- I might have it.

So what work have I done on Latin squares? Here's a list of my published or in progress papers in this field (click on a title to read the abstract).

1. M. J. L. Mendis and I. M. Wanless, Autoparatopisms of quasigroups and Latin squares, J. Combin. Designs to appear (201?).
2. J. Egan and I. M. Wanless, Enumeration of MOLS of small order, Math. Comp., 85 (2016) 799-824.
3. D. Donovan, K. Johnson and I. M. Wanless, Permanents and determinants of Latin squares, J. Combin. Designs 24 (2016), 132-148.
4. M. J. L. Mendis and I. M. Wanless, Latin squares with a unique intercalate, J. Combin. Designs 24 (2016), 279-293.
5. B. D. McKay, I. M. Wanless and X. Zhang, The order of automorphisms of quasigroups, J. Combin. Designs 23 (2015), 275-288.
6. J. Browning, P. J. Cameron and I. M. Wanless, Bounds on the number of small Latin subsquares, J. Combin. Theory Ser. A 124 (2014), 41-56.
7. T. Grubman and I. M. Wanless, Growth rate of canonical and minimal group embeddings of spherical latin trades, J. Combin. Theory Ser. A 123 (2014), 57-72.
8. N. J. Cavenagh, J. Kuhl and I. M. Wanless, Longest partial transversals in plexes, Ann. Comb. 18 (2014), 419-428.
9. I. M. Wanless and X. Zhang, On the existence of retransmission permutation arrays, Disc. Appl. Math., 161 (2013), 2772-2777.
10. I. M. Wanless and X. Zhang, Transversals of latin squares and covering radius of set of permutations, European J. Combin., 34, (2013) 1130-1143.
11. J. Browning, D. S. Stones and I. M. Wanless, Bounds on the number of autotopisms and subsquares of a Latin square, Combinatorica, 33, (2013) 11-22.
12. D. Bryant, N. J. Cavenagh, B. Maenhaut, K. Pula and I. M. Wanless, Non-extendible latin cuboids, SIAM J. Discrete Math. 26, (2012) 239-249.
13. J. Barát and I. M. Wanless, A cube dismantling problem related to bootstrap percolation, J. Stat. Phys., 149, (2012) 754-770.
14. J. Egan and I. M. Wanless, Latin squares with restricted transversals, J. Combin. Designs, 20 (2012), 124-141. (Click here for data)
15. D. S. Stones, P. Vojtechovský and I. M. Wanless, Cycle structure of autotopisms of quasigroups and Latin squares, J. Combin. Designs, 20 (2012), 227-263.
16. D. S. Stones and I. M. Wanless, How not to prove the Alon-Tarsi conjecture, Nagoya Math. J. 205 (2012), 1-24.
17. D. S. Stones and I. M. Wanless, A congruence connecting latin rectangles and partial orthomorphisms, Ann. Comb., 16 (2012), 349-365.
18. P. Danziger, I. M. Wanless and B. S. Webb, Monogamous Latin Squares, J. Combin. Theory Ser. A 118 (2011), 796-807.
19. J. Egan and I. M. Wanless, Indivisible partitions of latin squares, J. Statist. Plann. Inference, 141 (2011) 402-417.
20. J. Browning, P. Vojtechovský and I. M. Wanless, Overlapping latin subsquares and full products, Comment. Math. Univ. Carolin., 51 (2010) 175-184.
21. N. J. Cavenagh and I. M. Wanless, On the number of transversals in Cayley tables of cyclic groups, Disc. Appl. Math. 158 (2010), 136-146.
22. D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Th. Ser. A 117 (2010), 204-215.
23. D. Bryant, J. Egan, B. Maenhaut and I. M. Wanless, Indivisible plexes in latin squares, Des. Codes Cryptogr. 52 (2009), 93-105.
24. N.J. Cavenagh and I.M. Wanless, Latin trades in groups defined on planar triangulations, J. Algebraic Combin., 30 (2009), 323-347.
25. J. A. Egan and I. M. Wanless, Latin squares with no small odd plexes, J. Combin. Designs, 16 (2008) 477-492.
26. N. J. Cavenagh, C. Greenhill and I. M. Wanless, The cycle structure of two rows in a random latin square, Random Structures Algorithms 33 (2008) 286-309.
27. B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM J. Discrete Math. 22, (2008) 719-736. (Click here for data)
28. I. M. Wanless, A computer enumeration of small latin trades, Australas. J. Combin. 39, (2007) 247-258. (Click here for data)
29. I. M. Wanless, Transversals in latin squares, Quasigroups Related Systems 15, (2007) 169-190.
30. B. M. Maenhaut, I. M. Wanless and B. S. Webb, Subsquare-free Latin Squares of odd order, Euro. J. Combin. 28, (2007) 322-336.
31. B. D. McKay, J. C. McLeod and I. M. Wanless, The number of transversals in a Latin square, Des. Codes Cryptogr., 40, (2006) 269-284. (Click here for data)
32. I. M. Wanless and B. S. Webb, The Existence of Latin Squares without Orthogonal Mates, Des. Codes Cryptogr., 40, (2006) 131-135.
33. D. Bryant, B. M. Maenhaut and I. M. Wanless, New families of atomic Latin squares and perfect one-factorisations, J. Combin. Theory A 113, (2006) 608-624.
34. B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combin. 9 (2005) 335-344.
35. I. M. Wanless, Atomic Latin squares based on cyclotomic orthomorphisms, Electron. J. Combin. 12 (2005) R22. (Click here for extra data)
36. I. M. Wanless and E. C. Ihrig, Symmetries that Latin squares inherit from 1-factorizations, J. Combin. Des., 13 (2005) 157-172.
37. I. M. Wanless, Cycle switches in Latin squares, Graphs Combin. 20 (2004), 545-570.
38. I. M. Wanless, A partial latin squares problem posed by Blackburn, Bull. Inst. Comb. Appl., 42 (2004), 76-80.
39. I. M. Wanless, Diagonally cyclic latin squares, Euro. J. Combin., 25 (2004), 393-413.
40. B. M. Maenhaut and I. M. Wanless, Atomic Latin squares of order eleven, J. Combin. Designs, 12 (2004), 12-34.
41. M. Vaughan-Lee and I. M. Wanless, Latin squares and the Hall-Paige conjecture, Bull. London Math. Soc., 35 (2003) 1-5.
42. D. Bryant, B. M. Maenhaut and I. M. Wanless, A family of perfect factorisations of complete bipartite graphs, J. Combin. Theory A 98 (2002) 328-342.
43. I. M. Wanless, A generalisation of transversals for Latin squares, Electron. J. Combin. 9 (2002) R12.
44. I. M. Wanless, Answers to questions by Denes on Latin power sets, Euro. J. Combin., 22 (2001) 1009-1020.
45. I. M. Wanless, On McLeish's construction for Latin squares without intercalates, Ars Combin., 58 (2001) 313-317.
46. I. M. Wanless, Latin squares with one subsquare, J. Combin. Des. 9 (2001) 128-146.
47. B. D. McKay and I. M. Wanless, Most Latin squares have many subsquares, J. Combin. Theory Ser. A, 86 (1999) 323-347.
48. I. M. Wanless, Permanents, matchings and Latin rectangles (thesis abstract), Bull. Austral. Math. Soc., 59 (1999) 169-170.
49. I. M. Wanless, Perfect factorisations of bipartite graphs and Latin squares without proper subrectangles, Electron. J. Combin. 6 (1999) R9.

I've also worked on algebraic problems on groups, loops and quasigroups (important algebraic structures whose Cayley tables are latin squares). My papers in this area are:

1. M. Kinyon and I. M. Wanless, Loops with exponent three in all isotopes, Internat. J. Algebra Comput., 25 (2015), 1159-1177.
2. P. Vojtechovský and I. M. Wanless, Closest multiplication tables of groups, J. Algebra, 353 (2012), 261-285.
3. A. Brouwer and I. M. Wanless, Universally noncommutative loops, Bull. Inst. Comb. Appl., 61 (2011), 113-115.
4. D. Bryant, M. Buchanan and I. M. Wanless, The spectrum for quasigroups with cyclic automorphisms and additional symmetries, Discrete Math. 309 (2009) 821-833.

You can think of Latin squares as two dimensional permutations. I have also worked on the more standard one dimensional permutations, giving rise to:
1. C. J. Shallue and I. M. Wanless, Permutation Polynomials and Orthomorphism Polynomials of Degree Six, Finite Fields Appl. 20 (2013), 84-92.
2. D. S. Stones and I. M. Wanless, Compound orthomorphisms of the cyclic group, Finite Fields Appl. 16 (2010), 277-289.
3. P. J. Cameron and I. M. Wanless, Covering radius for sets of permutations, Discrete Math. 293 (2005) 91-109.

I have also done some work on Latin rectangles, which is what you get if you chop a few rows off a Latin square. My principle interest here has been in the following problem: For a rectangle of given dimensions, what structure gives the most possible extensions to a rectangle with one more row. Note that this is equivalent to a problem of maximising the permanent and also to maximising perfect matchings in regular bipartite graphs. My papers on this question are listed in the next section.

## Permanents

The permanent of a matrix is like the determinant but without those fiddly signs (+/-) on each term. My papers on permanents are
1. D. Donovan, K. Johnson and I. M. Wanless, Permanents and determinants of Latin squares, J. Combin. Designs 24 (2016), 132-148.
2. G.-S. Cheon and I. M. Wanless, Some results towards the Dittert conjecture on permanents, Linear Algebra Appl., 436, (2012) 791-801.
3. K. Pula, S.-Z. Song and I. M. Wanless, Minimum permanents on two faces of the polytope of doubly stochastic matrices, Linear Algebra Appl., 434 (2011) 232-238.
4. I. M. Wanless, On the Brualdi-Liu conjecture for the even permanent, Electron. J. Linear Algebra 17 (2008), 284-286.
5. G.-S. Cheon and I. M. Wanless, An interpretation of the Dittert conjecture in terms of semi-matchings, Discrete Math., 307, (2007) 2501-2507.
6. I. M. Wanless, Permanents, Chapter 31 in Handbook of Linear Algebra (ed. L. Hogben), Chapman & Hall/CRC (2007).
7. I. M. Wanless, On Minc's sixth conjecture, Linear and Multilinear Algebra, 55 (2007) 57-63.
8. I. M. Wanless, Addendum to Schrijver's work on minimum permanents, Combinatorica, 26 (2006) 743-745.
9. I. M. Wanless, Permanents of matrices of signed ones, Linear and Multilinear Algebra, 53 (2005) 427-433.
10. G.-S. Cheon and I. M. Wanless, An update on Minc's survey of open problems involving permanents, Lin. Alg. Appl., 403 (2005) 314-342.
11. I. M. Wanless, A lower bound on the maximum permanent in Λnk, Linear Algebra Appl. 373 (2003), 153-167.
12. I. M. Wanless, The Holens-Dokovic conjecture on permanents fails, Linear Algebra Appl. 286 (1999) 273-285.
13. I. M. Wanless, Maximising the permanent and complementary permanent of (0,1) matrices with constant line sum, Discrete Math., 205 (1999) 191-205.
14. B. D. McKay and I. M. Wanless, Maximising the permanent of (0,1) matrices and the number of extensions of Latin rectangles, Electron. J. Combin. 5 (1998) R11.
15. I. M. Wanless, Permanents, matchings and Latin rectangles (thesis abstract), Bull. Austral. Math. Soc., 59 (1999) 169-170.
16. I. M. Wanless, Jerrum's multistorey carpark, Austral. Math. Soc. Gaz., 23 (1996) 193-197.

## Graph theory

A graph in this sense is a network of nodes and connections between nodes. I have studied a few different problem areas, principally (i) matchings and factorisations, (ii) asymptotic enumerations and random graphs, (iii) graph games.

I've worked hard at matchings and one-factorisations in regular bipartite graphs, which has resulted in a number of publications already listed in previous sections (because these problems can be expressed in terms of Permanents and Latin squares).

I have also sweated over asymptotic enumerations and random graphs, with these results:

1. M. Minchenko and I. M. Wanless, Spectral moments of regular graphs in terms of subgraph counts, Lin. Alg. Appl. 446 (2014), 166-176.
2. I. M. Wanless, Counting matchings and tree-like walks in regular graphs, Combin. Probab. Comput. 19 (2010), 463-480.
3. P. Lieby, B. D. McKay, J. C. McLeod and I. M. Wanless, Subgraphs of random k-edge-coloured k-regular graphs, Combin. Probab. Comput. 18 (2009), 533-549.
4. B. D. McKay, I. M. Wanless and N. C. Wormald, Asymptotic enumeration of graphs with a given upper bound on the maximum degree, Combin. Probab. Comput. 11 (2002) 373-392.
5. Z. C. Gao, I. M. Wanless and N. C. Wormald, Counting 5-connected planar triangulations, J. Graph Theory 38 (2001) 18-35.
6. I. M. Wanless and N. C. Wormald, Regular graphs with no homomorphisms onto cycles, J. Combin. Theory B 82 (2001) 155-160.
7. B. D. McKay, I. M. Wanless and N. C. Wormald, The asymptotic number of graphs with a restriction on the maximum degree, Electron. Notes Discrete Math. 5 (2000), 228-230.
While at Melbourne Uni I also worked with Stephen Taylor and Natashia Boland on an amplifier placement problem, which resulted in...

1. S. Taylor, I. M. Wanless and N. L. Boland, Distance domination and amplifier placement problems, Australas. J. Combin. 34 (2006) 117-136.
Finally, there's some `holiday' mathematics I've done on graph games.
1. I. M. Wanless, Path achievement games, Australas. J. Combin. 23 (2001) 9-18.
2. I. M. Wanless, The periodicity of graph games, Australas. J. Combin. 16 (1997) 113-123.
Before closing this section I'll give you a question that I'd really like solved. Suppose I have a k-regular bipartite graph on 2n vertices (where n>2k) and I know that among all such graphs my graph has the most 4-cycles. Does my graph necessarily contain a complete bipartite component?

## What else keeps me busy at work?

I am an editor-in-chief of the Electronic Journal of Combinatorics and on the editorial board of the Journal of Combinatorial Designs and a reviewer for Zentralblatt. From 2006-2009 and 2013-2014 I was the president of the Combinatorial Mathematics Society of Australasia. In 2011, I was chair of the organising committee for 35ACCMCC.

## Contact Information

Tel: +61 3 99054442
E-mail: Is of the standard form firstname.lastname@monash.edu
Snail-mail:
```
School of Mathematical Sciences
Monash University
Vic 3800 Australia
```