On the lightness of chordal 4-cycle in 1-planar graphs with high minimum degree

Abstract

A graph G is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. The family of 1-planar graphs with minimum vertex degree at least δ and minimum edge degree at least ε is denoted by P 1 δ (ε). In this paper, it is proved that every graph in P 1 7 (14) (resp. P 1 6 (13)) contains a copy of chordal 4-cycle with all vertices of degree at most 10 (resp. 12).

Drawing complete multipartite graphs on the plane with restrictions on crossings

“…Almost two decades later, Borodin [3,4] improved this bound to 6 and showed the sharpness of the new bound. Recently, more and more papers on the coloring problems of 1-planar graphs appear (see the introduction in [14] for details). However, compared to the well-established planar graphs, the class of 1-planar graphs is still litter explored.…”

More on the structure of plane graphs with prescribed degrees of vertices, faces, edges and dual edges

“…A partial dependence between δ(G) and g(G) is also known: if δ(G) ≥ 5, then g(G) ≤ 4 and g(G) = 3 for δ(G) ∈ {6, 7}, see [3]; however, for δ(G) ∈ {3, 4}, an upper bound for g(G) is still not known. Also, not much is known on the dependence of dg(G) (which is a vague analogue of w * (G) for non-embedded graphs) on w(G), g(G) and δ(G): so far, the only result is the one from [10] that if δ(G) ≥ 6 and w(G) ≥ 13, then dg(G) = 6.…”

Strongly light subgraphs in the 1-planar graphs with minimum degree 7

“…Each 1-planar graph with minimum degree 7 contains a K 4 with all vertices of degree at most 11. Theorem 2.2 (Zhang et al [11]). Each 1-planar graph with minimum degree 7 contains a 4-cycle…”