Publications

Refereed Journal Publications

  1. A. Burgess, P. Danziger, T. Traetta, On the Hamilton-Waterloo problem with odd cycle lengths Journal of Combinatorial Designs 26 (2) (2018) 51-83.

  2. A. Burgess, P. Danziger, T. Traetta, On the Hamilton-Waterloo Problem with cycle lengths of distinct parities Discrete Mathematics, 341 (6) (2018) 1636-1644

  3. P. Danziger, S. Park, Orthogonally Resolvable Matching Designs Discrete Mathematics 341 (3) (2018) 701-704.

  4. A. Burgess, P. Danziger, T. Traetta, On the Hamilton-Waterloo problem with odd orders Journal of Combinatorial Designs, 25 (6) (2017) 258-287

  5. P. Danziger, E. Medelsohn, T. Traetta, On the existence of unparalleled even cycle systems European J. Combin. 59 (2017) 11-22

  6. S. Akbari, A. Burgess, P. Danziger, E. Mendelsohn, Zero-Sum Flows for Steiner Triple Systems Discrete Mathematics, 340 (3) (2017) 416-425

  7. R. Julian R. Abel, Robert F. Bailey, Andrea C. Burgess , Peter Danziger, Eric Mendelsohn, On generalized Howell designs with block size three 81 (2) 365-391 (2016)

  8. D. Bryant, A. Burgess, P. Danziger, Decompositions of Complete Graphs into Bipartite 2-Regular Subgraphs Electr. J. Comb. 23 (2): P2.1 (2016)

  9. M. Buratti, P. Danziger, A cyclic solution for an infinite class of Hamilton-Waterloo problems Graphs and Combinatorics, 32 (2) (2016) 521-531

  10. A. Burgess, P. Danziger, E. Mendelsohn, B. Stevens, Orthogonally Resolvable Cycle Decompositions Journal of Combinatorial Designs, 23 (8) (2015) 328-351.

  11. D. Bryant, P. Danziger, W. Pettersson, Bipartite 2-factorizations of complete multipartite graphs Journal of Graph Theory, 78 (4) (2015) 287-294.

  12. P. Danziger, D. Horsley, B. Webb, Resolvabiltiy of infinite designs Journal of Combinatorial Theory, Series A, 23 (2014), 73-85.

  13. N. Francetic, P. Danziger, E. Mendelsohn, Group Divisible Covering Designs with Block Size 4: A Type of Covering Array with Row Limit Journal of Combinatorial Designs, 21 (2013) 311-341,

  14. D. Bryant, P. Danziger, M. Dean, On the Hamilton-Waterloo Problem for Bipartite 2-Factors J. Combin. Designs, 21 (2013) 60-80

  15. P. Danziger, E. Mendolsohn Colourings of Graphs by k-Matchings Quaderni di Matematica, Recent Results in Designs and Graphs: a Tribute to Lucia Gionfriddo, edited by Marco Buratti, Curt Lindner, Francesco Mazzocca and Nicola Melone 28 (2013) 315-328.

  16. J. Lobb, C. Colbourn, P. Danziger, B. Stevens, J. Torres-Jimenez, Cover Starters for Strength Two Covering Arrays Discrete Mathematics, 312 (2012) 943-956.

  17. P. Danziger, I. Wanless, B. Webb Monogamous Latin Squares Journal of Combinatorial Theory, Series A, 118 (3), (2011), 796-807

      Classically the study of Mutually Orthogonal Latin Squares (MOLS) has focused on the question of how many orthogonal mates it is possible to have for a given order. Recently Wanless and Webb and independently Evans have shown that it is almost always possible to have a Bachelor square, a Latin square which has no mate. In this paper we consider the question of whether there exists a Latin Square which has exactly one mate, called a Monogamous square, for each order. We prove the following result:

      Theorem For any positive integer n > 6 there exists a Monogamous Latin Square of order n, except possibly when n = 2p for some prime p > 11.

  18. D. Bryant, P. Danziger, On bipartite 2-factorisations of Kn - I and the Oberwolfach problem Journal of Graph Theory, 68 (1) (2011), 22-37.

      The Oberwolfach problem was posed by Ringel in the 1960s. It relates to specification of tournaments and specifically to balanced seating arrangments at round tables. In this article we provide a complete solution to the Oberwolfach problem in the case where there are an even number of seats at each table. We in fact prove a much more general result.

  19. P. Danziger, M. Greig, B. Stevens Geometrical constructions of class-uniformly resolvable structures Journal of Combinatorial Designs, 19 (5) (2011) 329-344

      In this paper we use arcs, ovals, and hyperovals to construct class-uniformly resolvable structures. Many of the structures come from finite geometries, but we also use arcs from non-geometric designs. These techniques are useful for finding CURDs with a variety of block sizes including many with block sizes 2 and 4. In addition, these constructions give the first systematic way of constructing infinite families of CURDs with three block sizes.

  20. P. Danziger, E. Mendelsohn, L. Moura, B. Stevens Covering arrays avoiding forbidden edges Theoretical Computer Science, 410 (2009) 5403-5414.

      Covering arrays can be used to detect the existence of faulty pairwise interactions between parameters or components in a software system. The generalization considered here applies to the situation in which some input combinations are invalid, a requirement quite common in software testing.

      In this paper, we study covering arrays avoiding forbidden edges (CAFE's), where certain pairwise interactions are forbidden while all others must be covered, and we aim to minimize the number of tests. We establish a theoretical framework for this problem, by providing connections to the edge clique covering problem, lower and upper bounds, complexity results and a recursive construction. We also give an algorithm for the case of binary alphabets.

  21. P. Danziger, G. Quattrocchi, B. Stevens, The Hamilton-Waterloo Problem for Cycle Sizes 3 and 4 Journal of Combinatorial Designs, 17 (4) (2009), 342-352.

      The Hamilton-Waterloo problem is a generalised variant of the Oberwolfach problem and seeks a resolvable decomposition of the complete graph Kn, or the complete graph minus a 1-factor as appropriate, into cycles such that each resolution class contains only cycles of specified sizes. We completely solve the case in which the resolution classes are either all 3-cycles or 4-cycles, with a few possible exceptions when n = 24 and 48.

  22. P. Danziger, E. Mendelsohn, G. Quattrocchi, Resolvable decompositions of Kn into the union of two 2-paths, ARS Combin, 93 (2009)

      We give necessary and sufficient conditions for a resolvable H-decomposition of λ Kn in the case where H is one of the 10 graphs obtained by the union of two paths of length 2.

  23. Peter Danziger, Salvatore Milici, Gaetano Quattrocchi, Minimum embedding of a P4-design into a balanced incomplete block design of index λ, Discrete Mathematics, 309 (2009), pp. 4861-4870.

      For every pair of positive integers v, λ, we determine the minimum value of w such that there exists a balanced incomplete block design of order v+w, index λ and block-size 4 embedding a P4-design of order v and index 1 (P4 denotes the path of length 3).

  24. Peter Danziger, J.H. Dinitz, Alan C.H. Ling, Maximum Uniformly Resolvable Designs with Block Sizes 2 and 4, Discrete Mathematics, 309 (2009), pp. 4716-4721.

      A resolvable pairwise balanced design with each parallel classs consisting of blocks which are all of the same size is called a uniformly resolvable design, a URD. The necessary condition for the existence of a URD with block sizes 2 and 4 is that v = 0 (mod 4). Obviously in a URD with blocks of size 2 and 4 one wishes to have the maximum number of resolution classes of blocks of size 4, these designs are called maximum uniformly resolvable designs or MURDs. In this paper we consider the case when v = 0 (mod 12) and prove that a MURD(12u) exists for all u ≥ 3 with the possible exception of u ∈ {7,9,10,11,13,14,17,19,22,31,34,38,43,46,47,82}.

  25. Peter Danziger, Chengmin Wang, Resolvable Modified Group Divisible Designs with Higher Index, Australasian Journal of Combinatorics, 41 (2008) p. 37.

      A resolvable modified group divisible design (RMGD) is a modified group divisible design whose blocks can be partitioned into parallel classes. We show that the necessary conditions for the existence of a 3-RMGDDλ of type gu, namely g ≥ 3, u ≥ 3, gu = 0 mod 3 and λ(g-1)(u-1) = 0 mod 2, are sufficient with the two exceptions of (g, u, λ) ∈ {(6,3,1), (3,6,1)}.

  26. P. Danziger, D. Heap, E. Mendelsohn, Hill Climbing to Pasch Valleys Journal of Combinatorial Designs, 15 (5) (2007), p 405-419.

      Exhaustive enumeration of Steiner Triple Systems is not feasible, due to the the combinatorial explosion of instances. The next-best hope is to quickly find a sample that is representative of isomorphism classes. Stinson's Hill-Climbing algorithm is widely used to produce "random" Steiner Triple Systems, and certainly finds a sample of systems quickly, but the sample is not uniformly distributed with respect to the isomorphism classes of STS with v ≤ 19, and in particular, we find that isomorphism classes with a large number of Pasch configurations are under-represented. No analysis of the non-uniformity of the distribution with respect to isomorphism classes or the intractability of obtaining a representative sample for v > 19 is known. We also exhibit a modification to hill-climbing that makes the sample if finds closer to the uniform distribution over isomorphism classes in return for a modest increase in running time.

  27. Peter Danziger, Alistar Graham, Eric Mendelsohn The chromatic spectrum of graph designs, Bulletin of the ICA, 50 (2007) 71--96

  28. P. Danziger, Y. Tang, C. Wang, Resolvable Modified Group Divisible Designs, Journal of Combinatorial Designs, 15 (2007) 2--14.

  29. P. Danziger, Peter Dukes, Terry Griggs, Eric Mendelsohn, On the intersection problem for Steiner triple systems of different orders, Graphs and Combinatorics, 22 3 (2006) 311--329.

  30. P. Danziger, E. Mendelsohn, G. Quattrocchi, On the Chromatic Index of Path Decompositions, Discrete Mathematics. 284 (2004) no. 1-3, 107--121.

  31. P. Danziger, B. Stevens, Class-Uniformly Resolvable Group Divisible Structures I: Resolvable Group Divisible Designs, Electronic Journal of Combinatorics, 11(1) #R23, 2004.

  32. P. Danziger, B. Stevens, Class-Uniformly Resolvable Group Divisible Structures II: Frames, Electronic Journal of Combinatorics, 11(1) #R24, 2004.

  33. P. Danziger, E. Mendelsohn Bicolour graphs of Steiner triple systems. Papers on the occasion of the 65th birthday of Alex Rosa. Discrete Math. 261 (2003), no. 1-3, 157--176.

  34. P. Danziger, B. Stevens, Class-Uniformly Resolvable Designs, Journal of Combinatorial Designs, 9 (2), (2001), p79.

  35. P. Danziger, T. Griggs, M. Grannell, A. Rosa, On the 2-parallel chromatic index of Steiner triple systems, Australasian Journal of Combinatorics 17 (1998) pp. 109-131.

  36. P. Danziger, E. Mendelsohn, Intercalates everywhere, Journal of the London Mathematical Society, Septses Volume, 245 (1997) pp. 69-88

  37. P. Danziger, Uniform Restricted Resovable Designs with r=3, ARS Combin. 46 (1997) pp. 161-176.

  38. P. Danziger, E. Mendelsohn, Uniformly Resolvable Designs, J. Combin. Math. and Combin. Comput. 21 (1996), pp. 65 - 83.

  39. P. Danziger, E. Mendelsohn, T. Griggs and M. Grannell, 5-Line Configurations, Utilitas Math. 49 (1996) pp. 153-159.

Books and Chapters in Books

  1. P. Danziger, E. Mendelsohn, L. Moura, B. Stevens Covering arrays avoiding forbidden edges p. 296-308, in Combinatorial Optimization and Applications, Yang, Boting; Du, Ding-Zhu; Wang, Cao An (Eds.), Series: Lecture Notes in Computer Science, Springer, 2008, ISBN: 978-3-540-85096-0

  2. Course Notes for Foundations of Mathematical Thought. Strictly speaking this is not a publication, but it is a 190 page book.

  3. P. Danziger, P. Rodney, Uniformly Resolvable Designs, in The Handbook of Combinatorial Designs, J. Dinitz, C. Colbourn eds, CRC Press 1st ed. 1996.

Technical Reports

  1. P. Danziger, P. Dukes., T.S. Griggs and E. Mendelsohn, Small cases of the intersection problem for Steiner triple systems of different orders, University of Toronto Department of Computer Science Technical Reports.

  2. P. Danziger, B. Stevens, HCURF4 of type 19t+1 for 1 ≤ t ≤ 100, Technical report, 2003.

  3. P. Danziger, B. Stevens, HCURF's of type 65t+1, Technical report, 2003.

Maintained by Peter Danziger, September 2009.