On an extremal problem in the class of bipartite 1-planar graphs

Abstract

A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true for unbalanced

An annotated bibliography on 1-planarity

“…Also, for all n ≥ 4, there exist examples showing that these bounds are tight [93]; Figure 4(c) shows one of these graphs. Czap et al [48] showed that the maximum number of edges of bipartite 1-planar graphs that are almost balanced is not significantly smaller than 3n − 8, while the same is not true for unbalanced ones. In particular, if the size of the smaller partite set is sublinear in n, then the number of edges is (2 + o(1))n, while the same is not true otherwise.…”

On the Sizes of Bipartite 1-Planar Graphs

“…Theorem 4 provides a better upper bound for |E(G)| than Theorem 1. The authors in [5] mentioned a question of Sopena [11]: How many edges we have to remove from the complete bipartite graph with given sizes of the partite sets to obtain a…”

“…Note that Karpov's upper bound on the size of a bipartite 1-planar graph is in terms of its vertex number. When the sizes of partite sets in a bipartite 1-planar graph are taken into account, Czap, Przyby lo andŠkrabul'áková [5] obtained another upper bound for its size (i.e., Corollary 2 in [5]). In this paper we obtain the following result which proves Conjecture 3. x y 6.…”

“…In this paper we obtain the following result which proves Conjecture 3. x y 6. The result in [5,Lemma 4] shows that the upper bound for |E(G)| in Theorem 4 is tight when x 3 and y 6x − 12. Also, if x < 1 3 (y + 4), 1-planar graph?…”

“…[5]). If G is a bipartite 1-planar graph with partite sets of sizes x and y, where2 x y, then |E(G)| 2|V (G)| + 6x − 16.…”