(a) [BB] Show that the graph is planar by drawing an isomorphic plane graph with straight edges. (b) [BB] Label the regions defined by your plane graph and list the edges which form the boundary of each region. (c) [BB] Verify that V-E+R=2, N ≤2 E, and E ≤3 V-6 (where, as defined in the proof of Corollary 14.1 .3, N is the sum of the numbers of edges on the boundaries of all regions). | Numerade

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.

Convince yourself by drawing several examples that every 4 -val... | Filo

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.

Give an example of (i) a non-planar graph that is not homeomorphic to K5 or K3,3; (ii) a non-planar graph that is not contractible to K5 or K3,3. Why does the existence of these graphs not contradict Theorems 4.2 and 4.3 ? | Numerade

Planar Graph Perfect Matching is in NC