In this work we present a way of accelerating a column generation algorithm for the linear minimum cost multicommodity flow problem. We use a new model that, besides the usual variables corresponding to flows on paths, has a polynomial number of extra variables, corresponding to flows on circuits. Those extra variables are explicitly considered in the restricted master problem, from the beginni...

Walton and Welsh proved that if a co-loopless regular matroid M does not have minor in {M(K(3,3)),M∗(K5)}, then admits nowhere zero 4-flow. Lai, Li Poon {M(K5),M∗(K5)}, We prove {M((P10)¯3 ),M∗(K5)}, 4-flow where (P10)¯3 is the graph obtained from Petersen P10by contracting 3 edges of perfect matching. As both M(K3,3) M(K5) are contractions M((P10)¯3), our result extends results Poon.

We introduce a new family of mixed graphical models that consists of graphs with possibly directed, undirected and bidirected edges but without directed cycles. Moreover, there can be up to three edges between any pair of nodes. The new family includes Richardson’s acyclic directed mixed graphs, as well as Andersson-Madigan-Perlman chain graphs. These features imply that no family of mixed grap...

There is consensus that modularity has many benefits from cost savings due to increased commonality to enabling a higher variety of products. Full modularity is, however, not always achievable. How engineering systems and products whose design is heavily influenced by technical constraints, such as weight or size limitations, tend to exhibit rather integral architectures is shown in this study....

In this paper, we introduce a family of graphs which is a generalization of zero-divisor graphs and compute an upper-bound for the diameter of such graphs. We also investigate their cycles and cores

In this paper we solve the following problems: (i) find two differential operators P and Q satisfying [P, Q] = P , where P flows according to the KP hierarchy ∂P/∂t n = [(P n/p) + , P ], with p := ord P ≥ 2; (ii) find a matrix integral representation for the associated τ-function. First we construct an infinite dimensional space W = span C ψ 0 (z), ψ 1 (z),. .. of functions of z ∈ C invariant u...

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease th...

for an arbitrary ring $r$, the zero-divisor graph of $r$, denoted by $gamma (r)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $r$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. it is well-known that for any commutative ring $r$, $gamma (r) cong gamma (t(r))$ where $t(r)$ is the (total) quotient ring of $r$. in this...