Given a finite group G of even order, which graphs Γ have a 1−factorization admitting G as automorphism group with a sharply transitive action on the vertex-set? Starting from this question, we prove some general results and develop an exhaustive analysis when Γ is a complete multipartite graph and G is cyclic.

Under the traveling wave transformation, Fornberg-Whitham equation is reduced to an ordinary differential equation whose general solution can be obtained using the factorization technique. Furthermore, we apply the change of the variable and complete discrimination system for polynomial to solve the corresponding integrals and obtain the classification of all single traveling wave solutions to ...

Building on work of the fourth author in [68], we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Preliminary version, April 22, 1999.

We study the relationship between codes and unambiguous automata inside a sofic system. We show that a recognizable set is a code in a sofic system if and only if a particular automaton associated to the set and the shift is unambiguous. We discuss an example of a finite complete code in a sofic system in connection with the factorization conjecture.

We study the representation number for various sparse graphs; in particular, we give an exact formula for graphs with a single edge and complete binary trees and an improved lower bound for the representation number of the hypercube. We also study the prime factorization of the representation number of graphs with a single edge.

We introduce the concept of firm classes of morphisms as basis for the axiomatic study of completions of objects in arbitrary categories. Results on objects injective with respect to given morphism classes are included. In a finitely well-complete category, firm classes are precisely the coessential first factors of morphism factorization structures.

We study the representation number for some special sparse graphs. For graphs with a single edge and for complete binary trees we give an exact formula, and for hypercubes we improve the known lower bound. We also study the prime factorization of the representation number of graphs with one edge.

Let H be a connected subgraph of graph G. An H-factor G is spanning whose components are isomorphic to H. Given set mutually non-isomorphic graphs, uniform H-factorization partition the edges into H-factors for some H∈H. In this article, we give complete solution existence problem (Ck,Pk+1)-factorizations Kn−I in case when k even.