We study the problem of online bipartite matching, where algorithms have to draw irrevocable matchings based on an incomplete bipartite graph. Specifically, we focus on algorithms that maximize number of matchings (i.e. graphs with weight 0 or 1). First, competitive ratios of a well-studied problem (only one side of the bipartite graph is incomplete) with adversarial, random order and iid model...

Bipartite permutation graphs have several nice characterizations in terms of vertex ordering. Besides, as AT-free graphs, they have a linear structure in the sense that any connected bipartite permutation graph has a dominating path. In the present paper, we elaborate the linear structure of bipartite permutation graphs by showing that any connected graph in the class can be stretched into a "p...

A graph G is k-ordered if for any sequence of k distinct vertices v1, v2, . . . , vk of G there exists a cycle in G containing these k vertices in the specified order. In 1997, Ng and Schultz posed the question of the existence of 4-ordered 3-regular graphs other than the complete graph K4 and the complete bipartite graph K3,3. In 2008, Meszaros solved the question by proving that the Petersen ...

Let G be a bipartite graph, with k j e(G). The zero-sum bipartite Ramsey number B(G; Z k) is the smallest integer t such that in every Z k-coloring of the edges of K t;t , there is a zero-sum mod k copy of G in K t;t. In this paper we give the rst proof which determines B(G; Z 2) for all possible bipartite graphs G. In fact, we prove a much more general result from which B(G; Z 2) can be deduce...

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The ...

Let u and v be any two distinct nodes of an undirected graph G, which is k-connected. For 1 ≤ w ≤ k, a w-container C(u, v) of a k-connected graph G is a set of w-disjoint paths joining u and v. A w-container C(u, v) of G is a w-container if it contains all the nodes of G. A graph G is w-connected if there exists a w-container between any two distinct nodes. A bipartite graph G is w-laceable if ...

Let denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graphG ∈ is said to be a forcing face ofG if the subgraph ofG obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal system introduced in Che and Chen [Forcing ...

In this article, we extend several algebraic graph analysis methods to bipartite networks. In various areas of science, engineering and commerce, many types of information can be represented as networks, and thus the discipline of network analysis plays an important role in these domains. A powerful and widespread class of network analysis methods is based on algebraic graph theory, i.e., repre...

We introduce a notion of bipartite minors and prove a bipartite analog of Wagner’s theorem: a bipartite graph is planar if and only if it does not contain K3,3 as a bipartite minor. Similarly, we provide a forbidden minor characterization for outerplanar graphs and forests. We then establish a recursive characterization of bipartite (2, 2)-Laman graphs — a certain family of graphs that contains...

A graph is called supermagic if there is a labeling of edges where the edges are labeled with consecutive distinct positive integers such that the sum of the labels of all edges incident with any vertex is constant. A graph G is called degree-magic if there is a labeling of the edges by integers 1, 2, ..., |E(G)| such that the sum of the labels of the edges incident with any vertex v is equal t...