We consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cut generated from a rational lattice-free polyto...

We consider a vertex model in d dimensions characterized by lines which run in a preferred direction. We show that this vertex model is soluble if the weights of vertices with intersecting lines are given by a free-fermion condition, and that a fugacity −1 is associated to each loop of lines. The solution is obtained by mapping the model into a dimer problem and by evaluating a Pfaffian. We als...

In this paper, we study the a-labeling number GO'. of a bipartite graph G.

The bipartite Star123-free graphs were introduced by V. Lozin in [1] to generalize some already known classes of bipartite graphs. In this paper, we extend to bipartite Star123-free graphs a linear time algorithm of J. L. Fouquet, V. Giakoumakis and J. M. Vanherpe for finding a maximum matching in bipartite Star123, P7-free graphs presented in [2]. Our algorithm is a solution of Lozin’s conject...

In this paper, various properties of particular type of Hamiltonian graph and it's edge-disjoint Hamiltonian circuits have been discussed. It has been found that the intersection graph obtained from Euler Diagram is not Hamiltonian. The graph H(3m + 7, 6m + 14) for m ? 1, which is planner, regular of degree four, non-bipartite but Hamiltonian graph , has perfect matching 4 with nonrepeated...

The maximum rank completion problem is the problem of, given a partial matrix (that is, a matrix where we are only given some of the entries), ll in the unknown entries in such a way as to maximize the rank. Applications include bipartite matching and matroid intersection for linearly represented matroids. We describe an algorithm that nds a maximum rank completion by perturbing an arbitrary co...

We define a new kind of crossing number which generalizes both the bipartite crossing number and the outerplanar crossing number. We calculate exact values of this crossing number for many complete bipartite graphs and also give a lower bound. 1 Preliminaries The bipartite crossing number of a bipartite graph G was defined by Watkins in [W] to be the minimum number of crossings over all biparti...

A p-list assignment L of a graph G assigns to each vertex v of G a set L(v) ⊆ {0, 1, . . . , p − 1} of permissible colors. We say G is L-(p, q)-colorable if G has a (p, q)-coloring h such that h(v) ∈ L(v) for each vertex v. The circular list chromatic number χc,l(G) of a graph G is the infimum of those real numbers t for which the following holds: For any p, q, for any p-list assignment L with ...

In this paper, our purpose is twofold. Firstly, the tensor andresiduum operations on $L-$nested systems are introduced under thecondition of complete residuated lattice. Then we show that$L-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. Thus the new representation theorem of$L-$subsets on complete re...

The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the fi...