How Many Holes Can an Unbordered Partial Word Contain? ⋆ F. Blanchet-Sadri 1 , Emily Allen 2 , Cameron Byrum 3 , and Robert Merca¸ s 4 1 University of North Carolina, Department of Computer Science, P.O. Box 26170, Greensboro, NC 27402–6170, USA blanchet@uncg.edu 2 Carnegie Mellon University, Department of Mathematical Sciences, 5032 Forbes Ave., Pittsburgh, PA 15289, USA 3 University of Mississippi, Department of Mathematics, P.O. Box 1848, University, MS 38677, USA 4 GRLMC, Universitat Rovira i Virgili, Pla¸ca Imperial T´arraco, 1, Tarragona, 43005, Spain robertmercas@gmail.com Abstract. Partial words are sequences over a finite alphabet that may have some undefined positions, or “holes,” that are denoted by ⋄’s. A nonempty partial word is called bordered if one of its proper prefixes is compatible with one of its suffixes (here ⋄ is compatible with every letter in the alphabet); it is called unbordered otherwise. In this paper, we investigate the problem of computing the maximum number of holes a partial word of a fixed length can have and still fail to be bordered. 1 Introduction Motivated by a practical problem in gene comparison, Berstel and Boasson intro- duced the notion of partial words, or sequences over a finite alphabet that may contain some “holes” denoted by ⋄’s [1]. For instance, a⋄bca⋄b is a partial word with two holes over the three-letter alphabet {a, b, c}. Several interesting com- binatorial properties of partial words have been investigated, and connections have been made with problems concerning primitive sets of integers, partitions of integers and their generalizations, vertex connectivity in graphs, etc [3]. An unbordered word is a word such that none of its proper prefixes is one of its suffixes. Unbordered partial words were defined in [2], and two types of borders were identified: simple and nonsimple. In this paper, we investigate the maximum number of holes an unbordered partial word of length n over a k-letter alphabet can have. The contents of our paper are as follows: In Section 2, we compute the max- imum number of holes a nonsimply bordered partial word of length n over a ⋆ This material is based upon work supported by the National Science Foundation under Grant No. DMS–0754154. We thank the referees of a preliminary version of this paper for their very valuable comments and suggestions. A World Wide Web server interface has been established at www.uncg.edu/cmp/research/countingpwords for automated use of the program. This work was done during the fourth author’s stay at the University of North Carolina at Greensboro.