Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an l × l grid minor is exponential in l. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A grid-like-minor of order l in a graph G is a set of paths in Gwhose intersection graph is bipart...

1 Stable 2-Pairs and (X; Y)-Intersection Graphs 2 Abstract Given two xed graphs X and Y , the (X; Y)-intersection graph of a graph G is a graph where 1. each vertex corresponds to a distinct induced subgraph in G that is iso-morphic to Y , and 2. two vertices are adjacent ii the intersection of their corresponding sub-graphs contains an induced subgraph isomorphic to X. This notion generalizes ...

An L-segment consists of a horizontal and a vertical straight line which form an L. In an L-embedding of a graph, each vertex is represented by an Lsegment, and two segments intersect each other if and only if the corresponding vertices are adjacent in the graph. If the corner of each L-segment in an Lembedding lies on a straight line, we call it a monotone L-embedding. In this paper we give a ...

We study the problem of finding pairwise vertex-disjoint copies ?-vertex cycle C? in randomly perturbed graph model, which is union a deterministic n-vertex G and binomial random G(n, p). For ? ? 3 we prove that asymptotically almost surely U p) contains min{?(G), [n/l]} cycles C?, provided p C log n/n for sufficiently large. Moreover, when ?(G) ?n with 0 ? ?/l not ‘close’ to complete bipartite...

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...

Let k, l,m, n, and μ be positive integers. A Zμ-scheme of valency (k, l) and order (m,n) is an m × n array (Sij) of subsets Sij ⊆ Zμ such that for each row and column one has ∑n j=1 |Sij | = k and ∑m i=1 |Sij | = l, respectively. Any such scheme is an algebraic equivalent of a (k, l)-semiregular bipartite voltage graph with n and m vertices in the bipartition sets and voltages coming from the c...

Many real-world networks display a natural bipartite structure. It is necessary and important to study the bipartite networks by using the bipartite structure of the data. Here we propose a modification of the clustering coefficient given by the fraction of cycles with size four in bipartite networks. Then we compare the two definitions in a special graph, and the results show that the modifica...