Under what circumstances can an undirected graph be represented by integer lattice points in Cartesian coordinates? Specifics:
% Each point on the graph is mapped to (x,y) on the Cartesian grid where both x and y are integers.
% Two points (x1,y1) and (x2,y2) on the Cartesian grid are "connected" if abs(x1-x2)<=1 and abs(y1-y2)<=1. In other words, each point has 8 neighbors.
% Two points on the Cartesian graph representation should be connected iff there is an edge between those two points on the graph.
Examples:
% K4: All points are connected to each other.
12 34
% K2,2: 1 and 2 both connect to both 3 and 4 but there are no other connections.
3 1 2 4
Since I can't find a lattice representation for K3,2 I'm guessing lattice-able graphs are a proper subset of planar graphs.
Same question for 3D lattice points.