Planarity and Genus of Sparse Random Bipartite Graphs

Beyond-Planarity: Density Results for Bipartite Graphs

Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly-planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations. Early research dates back to the 1960s (e.g., Avital and Hanani [14]) for extremal problems on geometric graphs, but is also related to graph drawing problems ...

A planarity criterion for cubic bipartite graphs

We prove that a simple finite bipartite cubic non-planar graph contains a clean subdivision of K3.3. Here a subdivision of K3,3 is defined 1o be clean if it can be obtained from K3,3 by subdividing any edge by an even number of vertices. The proof is constructive and gives rise to a polynomial-time algorithm. @ 1998 Elsevier Science B.V. All rights reserved

Bipartite decomposition of random graphs

Definition (Maximal size complete bipartite induced subgraph). β(G) := size of maximal complete bipartite induced subgraph of G. Definition (Minimal bipartite decomposion number). τ(G) := minimal number of complete edge disjoint covering bipartite subgraphs of G. Definition (Minimal nontrivial bipartite decomposion number). τ (G) := minimal number of complete edge disjoint covering nontrivial (...

Nonorientable Genus of Nearly Complete Bipartite Graphs

Groupies in Random Bipartite Graphs

A vertex of a graph G is called a groupie if its degree is not less than the arithmetic mean of the degrees of its neighbors. Some results concerning groupies have been obtained in deterministic graph theory; see e.g. [1, 5, 6]. Recently, Fernandez de la Vega and Tuza [3] investigate groupies in Erdős-Rényi random graphs G(n, p) and shows that the proportion of the vertices which are groupies i...