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A 3-regular planar graph should satisfy the following conditions. The graph should have the same degree 3 [hence the name 3-regular]for all vertices, It also must be possible to draw the graph G such that the edges of the graph intersect only at vertices. [ In other words, the edges should not cross each other]. Viewed 3k times 2 Let G be a 3-regular connected planar graph with a planar embedding where each face has degree either 4 or 6 and each vertex is incident with exactly one face of degree 4. Determine the number of vertices, edges and faces of degree 4 and 6. Every maximal planar graph is a least 3-connected. If a maximal planar graph has v vertices with v > 2, then it has precisely 3v − 6 edges and 2v − 4 faces. Apollonian networks are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles. Equivalently, they are the planar 3 .
and edges of a polyhedron is planar. The graphs for regular polyhedra are the flrst and third graphs in Figure 5 and the three graphs in Figure 3. † For a connected planar graph G drawn on a plane, let D1;D2;;Df denote the faces of G, and let e(Di) denote the number of edges on the boundary of Di, 1 • i • r. If one counts the edges. Regular graphs of degree at most 2 are easy to classify: A 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of a disjoint union of cycles and infinite chains. A 3-regular graph is known as a cubic graph. The cube is a 3-regular planar bipartite graph with 8 vertices. $\endgroup$ – Gerry Myerson Feb 3 '12 at $\begingroup$ Your try for (b) shows that one attempt to build a graph fails. It doesn't show that every attempt must fail. $\endgroup$ – Gerry Myerson Feb 3 '12 at

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