Let G be a regular, connected planar graph of order 24. If every vertex of G has order 3, how many regions are in a planar representation of G? Explain your answer.
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- Let u and v be distinct vertices in a connected graph G. There may be several connected subgraphs of G containing u and v. What is the minimum size of a connected subgraph of G containing u and v? Explain your answer.Help me with b and cLet G be a planar graph. Use Euler's Formula to prove that G must have a vertex of degree 5 or less. Hint: what if it didn't?
- Prove that for any two edges of a 2-connected graph, a cycle exists containing both of them.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?Prove that the two graphs below are isomorphic. (c) Figure 4: Two undirected graphs. Each graph has 6 vertices. The ver- tices in the first graph are arranged in two rows and 3 columns. From left to right, the vertices in the top row are 1, 2, and 3. From left to right, the vertices in the bottom row are 6, 5, and 4. Undirected edges, line segments, are between the following vertices: 1 and 2; 2 and 3; 1 and 5; 2 and 5; 5 and 3; 2 and 4; 3 and 6; 6 and 5; and 5 and 4. The vertices in the second graph are a through f. Vertices d, a, and c, are vertically inline. Vertices e, f, and b, are horizontally to the right of vertices d, a, and c, respectively. Undirected edges, line segments, are between the following vertices: a and d; a and c; a and e; a and b; d and b; a and f; e and f; c and f; and b and f.
- If G is a connected planar graph where e = 3v - 6, show that every region is triangular (has three boundary edges).Prove that the two graphs below are isomorphic Figure 4: Two undirected graphs. Each graph has 6 vertices. The vertices in the first graph are arranged in two rows and 3 columns. From left to right, the vertices in the top row are 1, 2, and 3. From left to right, the vertices in the bottom row are 6, 5, and 4. Undirected edges, line segments, are between the following vertices: 1 and 2; 2 and 3; 1 and 5; 2 and 5; 5 and 3; 2 and 4; 3 and 6; 6 and 5; and 5 and 4. The vertices in the second graph are a through f. Vertices d, a, and c, are vertically inline. Vertices e, f, and b, are horizontally to the right of vertices d, a, and c, respectively. Undirected edges, line segments, are between the following vertices: a and d; a and c; a and e; a and b; d and b; a and f; e and f; c and f; and b and f.Consider the following directed graph. b Identify the correct statement about the antisymmetric property of the relation represented by the given directed graph. (You must provide an answer before moving to the next part.)
- Let G be a simple connected graph. (a) Suppose that G has t blocks. Prove that there is an ordering (B1,B2,...,Bt) of the blocks of G such that, for each i ∈ {2,3,...,t}, the graphs Bi and B1 ∪B2∪···∪Bi−1 have exactly one vertex in common. (b) Prove that the chromatic number of G is the maximum of the chromatic number of each block of G.Consider the following graph: d Provide a topological ordering of the vertices in this graph if there is one. If the vertices have no such ordering enter none Use only lowercase letters.6. Determine whether the following pair of graphs is isomorphic. Exhibit an isomorphism ( show the mapping between vertices) or provide an argument that none exists. V1 V3 V5 V1 V2 V5 VA V3 V4 V2 G1 G2