Abstract
An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.
Highlights
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once
We focus on a subclass of 1-planar graphs, in particular, outer-1-planar graphs
A graph is said to be outer 1-planar if it has a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge; such a drawing is called an outer-1-plane graph and the outer region of the drawing is called the outer boundary of G
Summary
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. [NZ20] showed that each 1-planar graph of minimum degree at least 3 contains an edge of type (3, ≤ 23), (4, ≤ 11), (5, ≤ 9), (6, ≤ 8) or [7, 7] It is not clear whether the bound 23 and 11 in the first two types are sharp, and the authors conjectured that they may be improved to 20 and 10, respectively. Our result implies that each outer-1-planar graph of minimum degree at least 2 contains an edge of type (2, ≤ 7) or [3, 3], and each maximal outer-1-planar graph contains an edge of type (2, ≤ 5) or [3, 3], and all bounds are sharp.
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