### In memoriam Bruce A Anderson

From BICA (8) 1993

Bruce was born May 11, 1939, in Bismarck, North Dakota. He completed his B.A., M.S. and Ph.D. degrees at the University of Iowa. His supervisor was Steve Armentrout, a topologist in the tradition of R.L. Moore. In 1966 Bruce joined the Department of Mathematics at Arizona State University where he spent his entire career. When he arrived, the department had an active group of general topol-ogists, and Bruce continued his research and directed his only Ph.D. student in that area. Throughout his career he was an exponent of the R.L. Moore legacy. His training had helped him develop to an admirable degree the habits of thor-oughness and self-reliance that characterized all aspects of his work. Although he later abandoned work in topology, he continued to think geometrically about the combinatorial constructions in which he became interested. His notebooks and lectures were full of multi-colored illustrations that guided his thinking. He also continued his adherence to "Moore method" instruction as much as his teaching assignments allowed.

Bruce investigated the cardinalities of maximal families of mutually complementary topologies on a given set. When he asked what happens when the underlying set is finite, the direction of his research changed permanently. His 1973 paper [1] "Finite topologies and Hamiltonian paths" was the first of 43 papers in combinatorics, and the last on a question from topology. It began his interest in perfect 1-factorizations and his use of starter-adder techniques, both of which continued throughout his remaining work. He began extensive correspondence with others investigating similar problems, and became a regular participant in the Southeastern Conferences on Combinatorics, Graph Theory and Computing. His first such conference was the Fourth, in 1973; his note [2] in its Proceedings records his debt to conference participants who led him to the literature on perfect one-factorizations and related problems.

Anderson made extensive contributions to three problems. His first results in combinatorics led to the construction (see [5]) of perfect 1-factorization of complete graphs in certain cases. He began also (see [61) the study of automorphism groups of classes of perfect 1-factorizations, a line of investigation that was carried farther by Ed Ihrig (see, for example, [5]), whose results indicate the importance of the 1-factorizations of Anderson. The survey article [18] contains more information on these matters. He also contributed to the study of Howell designs and related arrays. The spectrum for Room squares was determined by 1973. Bruce played an important role in the corresponding question for Howell designs. The reader should consult [14] and the survey paper [16] for more on his contributions. Finally, Bruce worked extensively on sequencings of finite groups, pursuing for non-abelian groups a question settled completely in the abelian case by Basil Gordon. He established [3] that certain dihedral groups are sequenceable, and introduced [4] symmetric sequencings and even starters. Many of Anderson's results on sequencings and the somewhat more general 2-sequencings or terraces are discussed in the survey paper [16]. (References [7] and [8] below give complete citations for papers listed as "to appear" in its bibliography.) Self-orthogonal 2-sequencings, together with many of the techniques useful in the construction of Howell designs, have led ([13], [91) to self-orthogonal Hamiltonian path decom-positions of 2 K, the complete multigraph on n vertices in which each edge has multiplicity two, for infinitely many new values of n.

The work on sequencings continued until it was interrupted by Bruce's illness. Two papers with Ihrig, still to appear as of this writing, contain significant advances. In the first [10] they established that every group of odd order has a starter-translate 2-sequencing (or terrace). In the second [11] they showed that every solvable group with a unique element of order two (except for the quatemian group of order eight) has a symmetric sequencing, extending the corresponding result [12] for Hamiltonian groups. Bruce's plans included attempting to extend this result to nonsolvable groups, and to investigate the nature of the 1-factorizations of the complete graph that these symmetric sequencings provide.

Bruce Anderson had a strong dedication to teaching, and expected a lot of his students. He had a special interest in the development of mathematical talent, and in providing a sound basis for students wishing to pursue further study in math-ematics. He played a major role in the undergraduate education of students who have gone on to doctoral study in mathematics. In November of 1990, Bruce was recognized by the Golden Key National Honor Society for "contributing significantly to the promotion of academic excellence through teaching."

Within the Department of Mathematics, Bruce was a strong member of several important committees. His consistency, thoroughness and integrity, and his human concern for faculty and staff, earned him the high regard of his colleagues.

With his wife Mary, a statistician and Associate Professor of Engineering, and two sons, Tal and Trev, Bruce enjoyed a wonderful family life. The family participated avidly in golf and in youth and church activities. Their welfare was his first priority, and their company his greatest joy.

During his final weeks Bruce shared a passage, noted in The Mathematical Experience by Davis and Hersh, with members of his family and with friends. The following words of Kepler helped him express his faith and peacefulness, and his gratitude for his life's work:

I thank You, 0 Lord, our Creator, that thou hast permitted me to look at the beauty in thy work of creation; I exult in the works of thy hands. See, I have completed the work to which I felt called; I have earned interest from the talents that thou has given me. I have proclaimed the glory of thy works to the people who will read these demonstrations, to the extent that the limitations of my spirit would allow.

Note: The author wishes to thank Ed Ihrig and Doug Stinson for their comments on earlier versions of this article.

References

1. B.A. Anderson, "Finite Topologies and Hamiltonian Paths," J. Comb. The-ory B 14 (1973), 87-93.

2. B.A. Anderson, "A Perfectly Arranged Room Square," Proc. of 4th South-eastern Con!. on Comb., Graph Theory and Computing (1973),141-150.

3. B.A. Anderson, "Sequencings of Certain Dihedral Groups," Proc. of 6th Southeastern Conf. on Comb., Graph Theory and Computing, (1975),65-76.

4. B.A. Anderson, "Sequencings and Starters," Pacific J. Math. 64 (1976), 17-24.

5. B.A. Anderson, "Some Perfect 1-Factorizations," Proc. of 7th Southeast-ern Conf. on Comb., Graph Theory and Computing (1976), 79-91.

6. B.A. Anderson, "Symmetry Groups of Some Perfect 1-Factorizations of Complete Graphs," Discrete Math. 18 (1977), 227-234.

7. B.A. Anderson, "All Dicyclic Groups of Order at Least Twelve have Sym-metric Sequencings," Contemporary Mathematics (AMS) 111 (1990), 5-21.

8. B.A. Anderson, "A Product Theorem for 2-Sequencings," Discrete Math.87 (1991),221-236.

9. B.A. Anderson, "A Doubling Construction for Certain Self-Orthogonal 2-Sequencings," Congressus Numerantium 34 (1991), 61-95.

10. B.A. Anderson and E.C. Ihrig, "All Groups of Odd Order have Starter-Translate 2-Sequencings," Australasian Journal of Combinatorics, to ap-pear.

11. B.A. Anderson and E.C. Ihrig, "Every finite solvable group with a unique element of order two, except the quaternian group, has a symmetric Se-quencing," J. Combinatorial Designs 1(1992), to appear.

12. B.A. Anderson and P.A. Leonard, "Symmetric Sequencings of Finite Hamil-tonian Groups with a Unique Element of Order 2," Congressus Numeran-tium 65 (1988), 147-158.

13. B.A. Anderson andP.A. Leonard, "A Class of Self-Orthogonal 2-Sequencings," Designs, Codes and Cryptography 1 (1991), 149-181.

14. B.A. Anderson, P.J. Schellenberg and D.R. Stinson, "The Existence of Howell Designs of Even Side," J. Comb. Theory A 36 (19

15. J. Denes and A.D. Keedwell, "Sequenceable and R-Sequenceable Groups: Row Complete Latin Squares," in Latin Squares: New Developments in the Theory and Applications, J. Denes and A. D. Keedwell, editors, North-Holland, 1991.

16. J. H. Dinitz and D.R. Stinson, "Room Squares and Related Designs," in Contemporary Design Theory: A Collection of Surveys, J.H. Dinitz and D.R. Stinson, editors, John Wiley and Sons, 1992.

17. E.C. thrig, "The structure of the symmetry groups of perfect one-factor-izations of K2, J. Combin. Theory B 47(1989), 307-329.

18. E. Seah, "Perfect One-factorizations of the Complete Graph—A Survey," Bulletin of the Institute of Combinatorics and its Applications 1 (1991), 59-70.

Bruce A. Anderson

May 11, 1939—May 5, 1992

P.A. Leonard

Department of Mathematics, Arizona State University

Tempe, Arizona 85287-1804, USA

Bruce A. Anderson, a Foundation Fellow of the Institute of Combinatorics and its Applications, passed away on May 5, 1992, after an illness of two months. His death brought the loss of a valued contributor to combinatorial mathematics. This note is a brief summary of his life and work.Bruce was born May 11, 1939, in Bismarck, North Dakota. He completed his B.A., M.S. and Ph.D. degrees at the University of Iowa. His supervisor was Steve Armentrout, a topologist in the tradition of R.L. Moore. In 1966 Bruce joined the Department of Mathematics at Arizona State University where he spent his entire career. When he arrived, the department had an active group of general topol-ogists, and Bruce continued his research and directed his only Ph.D. student in that area. Throughout his career he was an exponent of the R.L. Moore legacy. His training had helped him develop to an admirable degree the habits of thor-oughness and self-reliance that characterized all aspects of his work. Although he later abandoned work in topology, he continued to think geometrically about the combinatorial constructions in which he became interested. His notebooks and lectures were full of multi-colored illustrations that guided his thinking. He also continued his adherence to "Moore method" instruction as much as his teaching assignments allowed.

Bruce investigated the cardinalities of maximal families of mutually complementary topologies on a given set. When he asked what happens when the underlying set is finite, the direction of his research changed permanently. His 1973 paper [1] "Finite topologies and Hamiltonian paths" was the first of 43 papers in combinatorics, and the last on a question from topology. It began his interest in perfect 1-factorizations and his use of starter-adder techniques, both of which continued throughout his remaining work. He began extensive correspondence with others investigating similar problems, and became a regular participant in the Southeastern Conferences on Combinatorics, Graph Theory and Computing. His first such conference was the Fourth, in 1973; his note [2] in its Proceedings records his debt to conference participants who led him to the literature on perfect one-factorizations and related problems.

Anderson made extensive contributions to three problems. His first results in combinatorics led to the construction (see [5]) of perfect 1-factorization of complete graphs in certain cases. He began also (see [61) the study of automorphism groups of classes of perfect 1-factorizations, a line of investigation that was carried farther by Ed Ihrig (see, for example, [5]), whose results indicate the importance of the 1-factorizations of Anderson. The survey article [18] contains more information on these matters. He also contributed to the study of Howell designs and related arrays. The spectrum for Room squares was determined by 1973. Bruce played an important role in the corresponding question for Howell designs. The reader should consult [14] and the survey paper [16] for more on his contributions. Finally, Bruce worked extensively on sequencings of finite groups, pursuing for non-abelian groups a question settled completely in the abelian case by Basil Gordon. He established [3] that certain dihedral groups are sequenceable, and introduced [4] symmetric sequencings and even starters. Many of Anderson's results on sequencings and the somewhat more general 2-sequencings or terraces are discussed in the survey paper [16]. (References [7] and [8] below give complete citations for papers listed as "to appear" in its bibliography.) Self-orthogonal 2-sequencings, together with many of the techniques useful in the construction of Howell designs, have led ([13], [91) to self-orthogonal Hamiltonian path decom-positions of 2 K, the complete multigraph on n vertices in which each edge has multiplicity two, for infinitely many new values of n.

The work on sequencings continued until it was interrupted by Bruce's illness. Two papers with Ihrig, still to appear as of this writing, contain significant advances. In the first [10] they established that every group of odd order has a starter-translate 2-sequencing (or terrace). In the second [11] they showed that every solvable group with a unique element of order two (except for the quatemian group of order eight) has a symmetric sequencing, extending the corresponding result [12] for Hamiltonian groups. Bruce's plans included attempting to extend this result to nonsolvable groups, and to investigate the nature of the 1-factorizations of the complete graph that these symmetric sequencings provide.

Bruce Anderson had a strong dedication to teaching, and expected a lot of his students. He had a special interest in the development of mathematical talent, and in providing a sound basis for students wishing to pursue further study in math-ematics. He played a major role in the undergraduate education of students who have gone on to doctoral study in mathematics. In November of 1990, Bruce was recognized by the Golden Key National Honor Society for "contributing significantly to the promotion of academic excellence through teaching."

Within the Department of Mathematics, Bruce was a strong member of several important committees. His consistency, thoroughness and integrity, and his human concern for faculty and staff, earned him the high regard of his colleagues.

With his wife Mary, a statistician and Associate Professor of Engineering, and two sons, Tal and Trev, Bruce enjoyed a wonderful family life. The family participated avidly in golf and in youth and church activities. Their welfare was his first priority, and their company his greatest joy.

During his final weeks Bruce shared a passage, noted in The Mathematical Experience by Davis and Hersh, with members of his family and with friends. The following words of Kepler helped him express his faith and peacefulness, and his gratitude for his life's work:

I thank You, 0 Lord, our Creator, that thou hast permitted me to look at the beauty in thy work of creation; I exult in the works of thy hands. See, I have completed the work to which I felt called; I have earned interest from the talents that thou has given me. I have proclaimed the glory of thy works to the people who will read these demonstrations, to the extent that the limitations of my spirit would allow.

Note: The author wishes to thank Ed Ihrig and Doug Stinson for their comments on earlier versions of this article.

References

1. B.A. Anderson, "Finite Topologies and Hamiltonian Paths," J. Comb. The-ory B 14 (1973), 87-93.

2. B.A. Anderson, "A Perfectly Arranged Room Square," Proc. of 4th South-eastern Con!. on Comb., Graph Theory and Computing (1973),141-150.

3. B.A. Anderson, "Sequencings of Certain Dihedral Groups," Proc. of 6th Southeastern Conf. on Comb., Graph Theory and Computing, (1975),65-76.

4. B.A. Anderson, "Sequencings and Starters," Pacific J. Math. 64 (1976), 17-24.

5. B.A. Anderson, "Some Perfect 1-Factorizations," Proc. of 7th Southeast-ern Conf. on Comb., Graph Theory and Computing (1976), 79-91.

6. B.A. Anderson, "Symmetry Groups of Some Perfect 1-Factorizations of Complete Graphs," Discrete Math. 18 (1977), 227-234.

7. B.A. Anderson, "All Dicyclic Groups of Order at Least Twelve have Sym-metric Sequencings," Contemporary Mathematics (AMS) 111 (1990), 5-21.

8. B.A. Anderson, "A Product Theorem for 2-Sequencings," Discrete Math.87 (1991),221-236.

9. B.A. Anderson, "A Doubling Construction for Certain Self-Orthogonal 2-Sequencings," Congressus Numerantium 34 (1991), 61-95.

10. B.A. Anderson and E.C. Ihrig, "All Groups of Odd Order have Starter-Translate 2-Sequencings," Australasian Journal of Combinatorics, to ap-pear.

11. B.A. Anderson and E.C. Ihrig, "Every finite solvable group with a unique element of order two, except the quaternian group, has a symmetric Se-quencing," J. Combinatorial Designs 1(1992), to appear.

12. B.A. Anderson and P.A. Leonard, "Symmetric Sequencings of Finite Hamil-tonian Groups with a Unique Element of Order 2," Congressus Numeran-tium 65 (1988), 147-158.

13. B.A. Anderson andP.A. Leonard, "A Class of Self-Orthogonal 2-Sequencings," Designs, Codes and Cryptography 1 (1991), 149-181.

14. B.A. Anderson, P.J. Schellenberg and D.R. Stinson, "The Existence of Howell Designs of Even Side," J. Comb. Theory A 36 (19

15. J. Denes and A.D. Keedwell, "Sequenceable and R-Sequenceable Groups: Row Complete Latin Squares," in Latin Squares: New Developments in the Theory and Applications, J. Denes and A. D. Keedwell, editors, North-Holland, 1991.

16. J. H. Dinitz and D.R. Stinson, "Room Squares and Related Designs," in Contemporary Design Theory: A Collection of Surveys, J.H. Dinitz and D.R. Stinson, editors, John Wiley and Sons, 1992.

17. E.C. thrig, "The structure of the symmetry groups of perfect one-factor-izations of K2, J. Combin. Theory B 47(1989), 307-329.

18. E. Seah, "Perfect One-factorizations of the Complete Graph—A Survey," Bulletin of the Institute of Combinatorics and its Applications 1 (1991), 59-70.

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