For a set of non-negative integers L, the L-intersection number of a graph is the smallest number l for which there is an assignment of subsets Av ⊆ {1, . . . , l} to vertices v, such that every two vertices u, v are adjacent if and only if |Au ∩ Av| ∈ L. The bipartite L-intersection number is defined similarly when the conditions are considered only for the vertices in different parts. In this...

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 dimension of a bipartite graph with respect to a type L is the smallest number t for which it is possible to assign sets Ax ⊆ {1, . . . , t} of labels to vertices x so that any two vertices x and y from different parts are adjacent if and only if |Ax ∩Ay| ∈ L. The weight of such a representation is the sum ∑x |Ax| over all vertices x. We exhibit explicit bipartite n×n graphs wh...

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 spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph Γ is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the sp...

Edge-intersection graphs of paths in grids are graphs that can be represented with vertices as paths in grids and edges between the vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was ...

Tolerance graphs have been extensively studied since their introduction in 1982 [9], due to their interesting structure and their numerous applications, as they generalize both interval and permutation graphs in a natural way. It has been conjectured in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs [10]. The conjecture has been proved...

This paper shows that with B = {1, 2, . . . , n}, the smallest k such that (B ×B)− {(j, j) | j ∈ B} = k ⋃ i=1 (Ci ×Di) is s(n), where s(n) is the smallest integer k such that n 6 ( k b k 2 c ) . This provides a simple set-based formulation and a new proof of a result for boolean ranks [2] and biclique covering of bipartite graphs [1, 5], making these intricate results more accessible.

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideal...

We study the online matroid intersection problem, which is related to the well-studied online bipartite matching problem in the vertex arrival model. For two matroids M1 and M2 defined on the same ground set E, the problem is to design an algorithm that constructs a large common independent set in an online fashion. The algorithm is presented with the ground set elements one-by-one in a uniform...