Abstract
A plane graph is called alternating if all adjacent vertices have different degrees, and all neighboring faces as well. Alternating plane graphs were introduced in 2008. This paper presents the previous research on alternating plane graphs. There are two smallest alternating plane graphs, having 17 vertices and 17 faces each. There is no alternating plane graph with 18 vertices, but alternating plane graphs exist for all cardinalities from 19 on. From a small set of initial building blocks, alternating plane graphs can be constructed for all large cardinalities. Many of the small alternating plane graphs have been found with extensive computer help. Theoretical results on alternating plane graphs are included where all degrees have to be from the set {3,4,5}. In addition, several classes of “weak alternating plane graphs” (with vertices of degree 2) are presented.
Highlights
The concept of alternating plane graphs was introduced by I
This paper presents the previous research on alternating plane graphs
Many of the small alternating plane graphs have been found with extensive computer help
Summary
The concept of alternating plane graphs was introduced by I. Scherer’s squarings of a square, in particular the nice symmetric one where 21 small squares exactly fill a square of side length 16 in such a way that no two squares with the same side length join an edge or a vertex (see Figure 1) The 21-solution is the second smallest such object This concept of alternating tilings formed the inspiration for the definition of alternating plane graphs.
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