SOLVED:Draw a planar representation of each graph.

SOLVED: Consider a planar drawing of a graph all whose faces, includingthe outer one, are triangular (i.e., have 3 vertices). To eachvertex we arbitrarily assign one of the labels 1, 2, 3. Prove thatthere are an even number of faces whose vertices get all threelabels.

If G is a planar graph with a vertices, ecdges, f taces and k ... | Filo

Show that any minor of a planar graph is planar. Deduce that a graph is planar if and only if it is the minor of a grid. (Grids are defined in Chapter 12.3.) | Numerade

9.7 Give an example of each of the following or explain why no such example exists.(a) a planar graph of order 4.(b) a nonplanar graph of order 4.(c) a nonplanar graph of order 6 that contains neither K5 nor K3,3 as a subgraph.(d) a plane graph having 5 vertices, 10 edges and 7 regions.(e) a planar graph of order n ≥ 3 and size m with m = 3n - 6.(f) a nonplanar graph of order n ≥ 3 and size m with m = 3n - 6.

A planar graph G is said to be completely regular if the degrees of all vertices of G are equal and every region is bounded by the same number of edges. The graphs in Figs. 2-20(a) and 2-21(b) are completely regular, for example. Show that there are only five possible simple completely regular planar graphs, excluding the trivial graphs with degree ≤2. Sketch them. | Numerade

Show that every planar graph G can be colored using six or fewer colors. [Hint: Use mathematical induction on the number of vertices of the graph. Apply Corollary 2 of Section 10.7 to find a vertex v with deg(v) ≤5 . Con- sider the subgraph of G obtained by deleting v and all edges incident with it.] | Numerade

Determine whether the given graph is planar. If so, draw it so that no edges cross. | Numerade