We prove that it is NP-complete to decide whether a given string can be factored into palindromes that are each unique in the factorization.

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A K1,k-factorization of Km,n is a set of edge-disjoint K1,k-factors of Km,n which partition the set of edges of Km,n. When k is a prime number p, Wang [Discrete Math. 126 (1994)] investigated the K1,p-factorization of Km,n and gave a sufficient condition for such a factorization to exist. Du [Dis...

We show that a necessary and sufficient condition for the existence of a P2k-factorization of the complete bipartitemultigraph )"Km,n is m = n == 0 (mod k(2k l)/d), where d = gcd()", 2k 1).

We describe the complete factorization of the tenth Fermat number F10 by the elliptic curve method (ECM). F10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations

An automorphism group G of a 1-factorization of the complete multipartite graph Km×n consists of permutations of the vertices of the graph mapping factors to factors. In this paper, we give a complete answer to the existence problem of a 1-factorization of Km×n admitting a cyclic or dihedral group acting sharply transitively on the vertices of the graph.

Acknowledgments I would like to begin by thanking my advisor, Professor Alexander Diesl, for his encouragement and excitement for my research. He has always remained supportive and interested in my work, and his many questions have kept me motivated and excited about my project. I would like to thank him as well for being understanding about my silly questions, helping me piece apart my convolu...