P 4k−1-factorization of bipartite multigraphs

Abstract

Let λKm,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A Pv-factorization of λKm,n is a set of edge-disjoint Pv-factors of λKm,n which partition the set of edges of λKm,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a Pv-factorization of λKm,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k − 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P 4k−1 -factorization of λKm,n is (1) (2k − 1)m 2kn, (2) (2k − 1)n 2km, (3) m + n ≡ 0 (mod 4k − 1), (4) λ(4k − 1)mn/[2(2k − 1)(m + n)] is an integer.

The spectrum of path factorization of bipartite multigraphs

“…Very recently, we [10] proved that the conjecture is true when v = 4k − 1 (Theorem 3 in [10]). Theorem 3 [10] .…”

“…For our main result, we only need the following direct constructions. The first construction comes from [10].…”

The Research on Milling Temperature and Heat Deformation of Tool in Virtual Numerical Control Simulation

“…ξwill decrease when f increases, so the unit cutting force and unit cutting power decrease with the increasing of f, and the heat per unit volume of removing metal decrease. At the same time, the chip will be more thick, the heat capacity of chip will increase, and the heat taken away by chip will increase after f increases [7], so the rising of temperature in the cutting zone is not significant.…”