let G be a connected plane k regular graph in which each face is bounded by a cycle of length l show that 1/k + 1/l > 1/2
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let G be a connected plane k regular graph in which each face is bounded by a cycle of length l show that 1/k + 1/l > 1/2
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- Let G be a planar graph on 12 vertices with 24 edges in which all faces are bounded by cycles of length 3 or 4. How many triangles does G contain?Find all r-regular graphs of order n for a convenient selection of r, n.Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane graph with n vertices and girth g.
- Prove that every connected planar graph with less than 12 vertices has a vertex of degree at most 4. [Hint: Assume that every vertex has degree at least 5 to obtain a lower bound on e (together with the upper bound on e in the corollary) that implies v ≥ 12.]How many nonisomoprhic connected planar graphs (V, E) are there with v = 6 and e = 10 assuming that each vertex has degree ≥ 3 ?Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to 2v-4, where v= number of vertices and f= faces of H. Show that for any integer v greater than or qual to 3, there exists a connected planar graph H that has v vertices and 2v-4 faces.