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Dr 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 a senior research fellow at Monash University in Melbourne.

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.

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!

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. D. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, (submitted).
  2. J. A. Egan and I. M. Wanless, Latin squares with no small odd plexes, J. Combin. Designs, to appear.
  3. D. Bryant, M. Buchanan and I. M. Wanless, The spectrum for quasigroups with cyclic automorphisms and additional symmetries, Discrete Math. (to appear).
  4. N. J. Cavenagh, C. Greenhill and I. M. Wanless, The cycle structure of two rows in a random latin square, Random Structures Algorithms (to appear).
  5. 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)
  6. I. M. Wanless, A computer enumeration of small latin trades, Australas. J. Combin. 39, (2007) 247-258. (Click here for data)
  7. I. M. Wanless, Transversals in latin squares, Quasigroups Related Systems 15, (2007) 169-190.
  8. B. M. Maenhaut, I. M. Wanless and B. S. Webb, Subsquare-free Latin Squares of odd order, Euro. J. Combin. 28, (2007) 322-336.
  9. 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.
  10. I. M. Wanless and B. S. Webb, The Existence of Latin Squares without Orthogonal Mates, Des. Codes Cryptogr., 40, (2006) 131-135.
  11. 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.
  12. B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combin. 9 (2005) 335-344.
  13. I. M. Wanless, Atomic Latin squares based on cyclotomic orthomorphisms, Electron. J. Combin. 12 (2005) R22. (Click here for extra data)
  14. I. M. Wanless and E. C. Ihrig, Symmetries that Latin squares inherit from 1-factorizations, J. Combin. Des., 13 (2005) 157-172.
  15. I. M. Wanless, Cycle switches in Latin squares, Graphs Combin. 20 (2004), 545-570.
  16. I. M. Wanless, A partial latin squares problem posed by Blackburn, Bull. Inst. Comb. Appl., 42 (2004), 76-80.
  17. I. M. Wanless, Diagonally cyclic latin squares, Euro. J. Combin., 25 (2004), 393-413.
  18. B. M. Maenhaut and I. M. Wanless, Atomic Latin squares of order eleven, J. Combin. Designs, 12 (2004), 12-34.
  19. M. Vaughan-Lee and I. M. Wanless, Latin squares and the Hall-Paige conjecture, Bull. London Math. Soc., 35 (2003) 1-5.
  20. 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.
  21. I. M. Wanless, A generalisation of transversals for Latin squares, Electron. J. Combin. 9 (2002) R12.
  22. I. M. Wanless, Answers to questions by Denes on Latin power sets, Euro. J. Combin., 22 (2001) 1009-1020.
  23. I. M. Wanless, On McLeish's construction for Latin squares without intercalates, Ars Combin., 58 (2001) 313-317.
  24. I. M. Wanless, Latin squares with one subsquare, J. Combin. Des. 9 (2001) 128-146.
  25. B. D. McKay and I. M. Wanless, Most Latin squares have many subsquares, J. Combin. Theory Ser. A, 86 (1999) 323-347.
  26. I. M. Wanless, Permanents, matchings and Latin rectangles (thesis abstract), Bull. Austral. Math. Soc., 59 (1999) 169-170.
  27. I. M. Wanless, Perfect factorisations of bipartite graphs and Latin squares without proper subrectangles, Electron. J. Combin. 6 (1999) R9.

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. I. M. Wanless, On the Brualdi-Liu conjecture for the even permanent, Electron. J. Linear Algebra 17 (2008), 284-286.
  2. G.-S. Cheon and I. M. Wanless, An interpretation of the Dittert conjecture in terms of semi-matchings, Discrete Math., 307, (2007) 2501-2507.
  3. I. M. Wanless, Permanents, Chapter 31 in Handbook of Linear Algebra (ed. L. Hogben), Chapman & Hall/CRC (2007).
  4. I. M. Wanless, On Minc's sixth conjecture, Linear and Multilinear Algebra, 55 (2007) 57-63.
  5. I. M. Wanless, Addendum to Schrijver's work on minimum permanents, Combinatorica, 26 (2006) 743-745.
  6. I. M. Wanless, Permanents of matrices of signed ones, Linear and Multilinear Algebra, 53 (2005) 427-433.
  7. 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.
  8. I. M. Wanless, A lower bound on the maximum permanent in Λnk, Linear Algebra Appl. 373 (2003), 153-167.
  9. I. M. Wanless, The Holens-Dokovic conjecture on permanents fails, Linear Algebra Appl. 286 (1999) 273-285.
  10. I. M. Wanless, Maximising the permanent and complementary permanent of (0,1) matrices with constant line sum, Discrete Math., 205 (1999) 191-205.
  11. 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.
  12. I. M. Wanless, Permanents, matchings and Latin rectangles (thesis abstract), Bull. Austral. Math. Soc., 59 (1999) 169-170.
  13. 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. P. Lieby, B. D. McKay, J. C. McLeod and I. M. Wanless, Subgraphs of random k-edge-coloured k-regular graphs, submitted.
  2. 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.
  3. Z. C. Gao, I. M. Wanless and N. C. Wormald, Counting 5-connected planar triangulations, J. Graph Theory 38 (2001) 18-35.
  4. I. M. Wanless and N. C. Wormald, Regular graphs with no homomorphisms onto cycles, J. Combin. Theory B 82 (2001) 155-160.
  5. 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 in Melbourne 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 a managing editor of the Electronic Journal of Combinatorics, the president of the Combinatorial Mathematics Society of Australasia and a reviewer for Zentralblatt.

If you're looking for a great combinatorics conference come to 4ICC in Auckland, 15-19 December 2008.

Contact Information

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

Last updated: Nov 2007