The structure of 1-planar graphs

Light 3-Cycles in 1-Planar Graphs With Degree Restrictions

“…The notion of 1-planarity was introduced by Ringel [14], who proved that each 1-planar graph is vertex 7-colorable. This is the first result on the colorings of 1-planar graphs, and from then on, many authors started to investigate the coloring problems (see [1,3,4,6,15,16,17,21,23,24]) and the structural properties (see [5,8,9,10,11,13,18,19,20,22]) of 1-planar graphs. One of possible approaches in the study of local graph structures can be formalized in the following way (see [12]):…”

“…In 2007, Fabrici and Madaras [8] completely determined the set of light graphs in the family P 1 4 (8); they are P 1 , P 2 and P 3 . Recently, the set of light 512 X. ZHANG graphs in the family P 1 5 (10) is also completely determined (see [8,7,19]); they are P 1 , P 2 , P 3 , P 4 and S 3 , but L(P 1 6 (12)) and L(P 1 7 (14)) are still undetermined.…”

“…In this paper, we prove that the 3-cycle is light in the family of 1-planar graphs with minimum vertex degree at least 5 and minimum edge degree at least 12. This generates a known result of Fabrici and Madaras [8]. …”

(1,λ)-Embedded GRAPHS AND THE ACYCLIC EDGE CHOOSABILITY

“…Actually, a (1, 2)-embedded graph is a 1-planar graph. It is shown in many papers such as [6] that e(G) ≤ 4v(G) − 8 for every 1-planar graph G. Whereafter, to determine whether the number of edges in the class of (1, λ)-embedded graphs is linear or not linear in the number of vertices for every λ ≤ 2 might be interesting.…”

A Linear-Time Algorithm for Testing Outer-1-Planarity

“…Pach and Toth [5] prove that a 1-planar graph with n vertices has at most 4n − 8 edges, which is a tight upper bound. There are a number of structural results on 1-planar graphs [6], and maximal 1-planar embeddings [7] (a 1-planar embedding of a graph G is maximal, if no edge can be added without violating the 1-planarity of G).…”