Degree Sequences and the Existence of k-Factors

We consider sufficient conditions for a degree sequence π to be forcibly k-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially k-factor graphical. We first give a theorem for π to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most β ≥ 0. These theorems are equal in strength to Chvátal’s well-known hamiltonian theorem, i.e., the best monotone degree condition for hamiltonicity. We then give an equally strong theorem for π to be forcibly 2-factor graphical. Unfortunately, the number of nonredundant conditions that must be checked increases significantly in moving from k = 1 to k = 2, and we conjecture that the number of nonredundant conditions in a best monotone theorem for a k-factor will increase superpolynomially in k. This suggests the desirability of finding a theorem for π to be forcibly k-factor graphical whose algorithmic complexity grows more slowly. In the final section, we present such a theorem for any k ≥ 2, based on Tutte’s well-known factor theorem. D. Bauer Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, U.S.A. E-mail: [email protected] H.J. Broersma Department of Computer Science, Durham University, South Road, Durham DH1 3LE, U.K. (Current address: Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. E-mail: [email protected]) J. van den Heuvel Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, U.K. E-mail: [email protected] N. Kahl Department of Mathematics and Computer Science, Seton Hall University, South Orange, NJ 07079, U.S.A. E-mail: [email protected] E. Schmeichel Department of Mathematics, San José State University, San José, CA 95192, U.S.A. E-mail: [email protected]