4-Labelings and Grid Embeddings of Plane Quadrangulations
Finding aesthetic drawings of planar graphs is a main issue in graph drawing. Of special interest are rectangle of influence drawings.The graphs considered here are quadrangulations, that is, planar graphs all whose faces have degree four.We show that each quadrangulation on n vertices has a closed rectangle of influence drawing on the (n - 2) × (n - 2) grid. Biedl, Bretscher and Meijer  proved that every planar graph on n vertices without separating triangle has a closed rectangle of influence drawing on the (n - 1) × (n - 1) grid.Our method, which is completely different from that of , is in analogy to Schnyder’s algorithm for embedding triangulations on an integer grid  and gives a simple algorithm.
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