For n even, a factorization of a complete graph Kn is a partition of the edges into n−1 perfect matchings, called the factors of the factorization. With respect to a factorization, a path is called rainbow if its edges are from distinct factors. A rainbow Hamiltonian path takes exactly one edge from each factor and is called orthogonal to the factorization. It is known that not all factorizatio...

Recently, Katre et al. introduced the concept of coprime index a graph. They asked to characterize graphs for which is same as clique number. In this paper, we partially solve problem. fact, prove that number and zero-divisor graph an ordered set ring Zpn coincide. Also, it proved annihilating ideal graphs, co-annihilating comaximal commutative rings can be realized specially constructed posets...

It is shown that the classical Infeld-Hull factorization method can be extended to coupled systems of second-order equations. A complete solution of the factorization equations in two dimensions is given and a partial enumeration of factorizable systems is made.