Quadratischplanare

Quadratischplanare, in graph theory often referred to as quadrangulations or plane quadrangulations, are planar graphs embedded in the plane such that every face, including the outer face, is bounded by a 4-cycle. In other words, each face is a quadrilateral. The term is common in German-language literature, while English sources frequently use the terms quadrangulation or quad-graph.

Key properties follow from the uniform face length. Such graphs are bipartite, since all cycles have even length. The dual graph of a quadrangulation is 4-regular. For a connected quadrangulation of the sphere, Euler’s formula implies tight relations between vertices V, edges E, and faces F: E = 2V − 4 and F = V − 2, so the average vertex degree is 4 − 8/V. These constraints mean small quadrangulations have lower average degree, approaching 4 as the graph grows.

Examples include the graph of a cube, which forms a planar quadrangulation with eight vertices, twelve edges, and six square faces. More generally, many grid-like graphs and mesh structures admit quadrangulations, making them useful in various applications.

Applications and related concepts: Quadrangulations play a central role in quadrilateral mesh generation for finite element analysis and computer graphics, where faces are desirable to be four-sided for quality and regularity. They are closely related to triangulations (via duality and construction techniques) and to the broader study of planar graphs and mesh topologies. See also planar graphs, dual graphs, and quadrangulation.

In summary, quadratischplanare graphs describe planar embeddings with exclusively quadrilateral faces, yielding characteristic bipartite structure and specific Euler-based relations that underpin their theory and applications.