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- combinatorial designs
- graph theory
- algebraic combinatorics
- finite geometries
- computational complexity
- coding theory
Current Project: Extremal Regular Graphs with Given Girth
The problem of finding smallest k-regular graphs with given girth g (called (k, g)-cages) was first discussed by Tutte in 1947. These graphs have been intensely studied after the publication of the basic works of Erdös and Sachs (in which they showed the existence of cages), and Hoffman and Singleton (in which they showed the nonexistence of certain Moore graphs). The study of cages has led to interesting applications of algebra and finite geometries to graph theory. The problem of finding cages is very difficult in general. Thus far, only a few cages are known and, furthermore, good upper bound for cages has not been found.
More recently, the concept of girth pair was introduced. In 1993, Dr. Zhang made a very important contribution to the field by completely determining the smallest number of vertices of a regular graph with given odd girth. A related problem is that of finding extremal regular graphs of given (odd) girth without one-factorization. The problem is very closely related to the well-known Four-Color Theorem and many other problems in graph theory. Dr. Zhang has recently made some significant progress on the problem.
- Tight Single-Change Covering Designs, Utilitas Mathematica, v. 47 (1995), 55-84. (with D.R. Preece, R.L. Constable, J.L. Yucas, W.D. Wallis, J.P. McSorley and N.C.K. Phillips).
- Pseudo-One-Factorizations of Regular Graphs of Odd Order-II: Products of Graphs, Discrete Mathematics, v. 128 (1994), 371-384.
- On the Iteration of Graph Labelings, Ars Combinatoria, v. 37 (1994), 75-85.
- Some New Bounds on Single-Change Covering Designs, SIAM J. Discrete Mathematics, v. 7 (1994), 166-171.
- Extremal Regular Graphs with Prescribed Odd Girth, J. of Combinatorial Theory, Series B, v. 60 (1994), 222-238.
- Ph.D. (math) Southern Illinois University, 1990
- M.S. (math) Southern Illinois University, 1987.
- B.S. (math) Northeast Normal University, 1983.
- At UAH since 1993.