A graph is called $1$-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let $G$ be a bipartite $1$-planar graph with $n$ ($nge 4$) vertices and $m$ edges. Karpov showed that
We prove that maximal possible sizes of bipartite $1$-planar graphs whose one partite set is much smaller than the other one tends towards $2n$ rather than $3n$. In particular, we prove that if the size of the smaller partite set is sublinear in $n$, then $|E|=(2+o(1))n$, ...
D. V. Karpov. An upper bound on the number of edges in an almost planar bipartite graph. J. Math. Sci., 196(6):737-746, 2014.D. V. Karpov. An upper bound on the number of edges in an almost planar bipartite graph. Journal of Mathematical Sciences, 196(6):737-746, 2014....
The complete bipartite graph (2-partite graph) with part sizes m and n is denoted by Km,n. The girth of G is the length of a shortest cycle in G and is denoted by gr(G). If G has no cycles, we define the girth of G to be infinite. A graph G is said to be planar if it...
The g-spectrum of a graph is the set of sizes of its g-convex sets. In this paper we consider two problems — counting g-convex sets in a graph, and determining when a graph has g-convex sets of every cardinality (such graphs are said to have the continuum property). We show that...
Predicting the interaction affinity between drugs and target proteins is crucial for rapid and accurate drug discovery and repositioning. Therefore, more accurate prediction of DTA has become a key area of research in the field of drug discovery and drug
property that the sizes of any two color classes differ by at most one? Previous Results. LIST COLORING is NP-complete, even for very restricted classes of graphs, such as complete bipartite graphs [JS97]. Jansen and Scheffler described a dynamic programming algorithmfor the problem that ru...
First, we recall the concept of DP-coloring of graphs introduced by Dvořák and Postle [4]. Definition 1.1 Let G be a graph. A cover of G is a pair (F,L), where F is a graph and L:V(G)→Pow(V(F)) is a function, with the following properties: • the sets L(u), u...
Thus, a cycle double cover of a graph on e edges has total size 2e. The minimum total size of a cycle cover of a bridgeless graph G is denoted by scc(G) (for shortest cycle cover). There are a number of interesting questions on shortest cycle covers of graphs. In 1995, Thomassen ...
maximal matching; partition into triangles; partition into isomorphic subgraphs; partition into Hamiltonian subgraphs; partition into forests; partition into cliques; partition into perfect matchings; two-stage maximum weight stochastic matching; covering by cliques; and covering by complete bipartite ...