As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekulé structures (1-factors). A bipartite graph G is called elementary if G is connected and every edge belongs to a 1-factor of G. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given. A peripheral face f of a plane elementary graph is reducible, if th...

A sparse subgraph G′ of G is called a matching sparsifier if the size or weight of matching in G′ is approximately equal to the size or weight of maximum matching in G. Recently, algorithms have been developed to find matching sparsifiers in a static bipartite graph. In this paper, we show that we can find matching sparsifier even in an incremental bipartite graph. This observation leads to fol...

A proper edge coloring of a graph G with colors 1, 2, 3, . . . is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graph is (3, 4)-biregular if all vertices in one part have degree 3 and all vertices in the other part have degree 4. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph G has a spanning subgraph...

Lately, a lot of research has been done on C -probe graphs. In this paper we focus on chordal-bipartite probe graphs. We prove a structural result that if B is a bipartite graph with no chordless cycle of length strictly greater than 6, then B is chordalbipartite probe if and only if a certain “enhanced” graph B∗ is a chordal-bipartite graph. This theorem is analogous to a result on interval pr...

We prove that a triangle-free graph G is a tolerance graph if and only if there exists a set of consecutively ordered stars that partition the edges ofG. Since tolerance graphs are weakly chordal, a tolerance graph is bipartite if and only if it is triangle-free.We, therefore, characterize those tolerance graphs that are also bipartite. We use this result to show that in general, the class of i...

Matching extendability is significant in graph theory and its applications. The basic notion in this direction is n-extendability introduced by Plummer in 1980. Motivated by the different natures of bipartite matchings and non-bipartite matchings, this paper investigates bipartite-matching extendable (BM-extendable) graphs. A graph G is said to be BM-extendable if every matching M which is a pe...

Bipartite graph tiling was studied by Zhao [7] who gave the best possible minimum degree conditions for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of Ks,s. Let s < t be fixed positive integers. Hladký and Schacht [3] gave minimum degree conditions for a balanced bipartite graph on 2m(s + t) vertices to contain m vertex disjoint copies of Ks,t. Their results w...

We give a method to construct cospectral graphs for the normalized Laplacian by a local modification in some graphs with special structure. Namely, under some simple assumptions, we can replace a small bipartite graph with a cospectral mate without changing the spectrum of the entire graph. We also consider a related result for swapping out biregular bipartite graphs for the matrix A + tD. We p...

In this paper we study when a bipartite graph is a covering of a non-bipartite graph. We give a characterization of all coverings in terms of factoring the covering map through the canonical double covering. We also consider regular bipartite coverings described in terms of voltage assignments. We give an algebraic characterization of such coverings involving the subgroup generated by voltages ...

Given a fixed bipartite graph H, we study the asymptotic speed of growth of the number of bipartite graphs on n vertices which do not contain an induced copy of H. Whenever H contains either a cycle or the bipartite complement of a cycle, the speed of growth is 2Ω(n 6 5 ). For every other bipartite graph except the path on seven vertices, we are able to find both upper and lower bounds of the f...