A graph G is called self-complementary if it isomorphic to its complement G. For a self-complementary graph G = (V, E), let |V| = n, then  (a) find |E|, and  (b) prove that G is connected

(Discrete Math-Graph Theory) Let G=(V,E₁∪E₂∪E₃) be a simple graph such that G₁=(V,E₁) is planar, G₂=(V,E₂) is a forest, and G₃=(V,E₃) is a matching. Prove G is 9-colourable, i.e. its chromatic number satisfies χ(G)≤9.

Can someone provide the proof for this? TIA

Let G be a graph of order n that is isomorphic to its complement G. How many edges does G have? Explain your answer. If a graph G has n vertices, all of which but one have odd degree, how many vertices of odd degree are there in G, the complement of G? Prove your answer.

6.) Give a proper coloring for the two graphs below. Then determine the chromatic number of each, Justifi your answer. √1 V3 G₁ V4 V5 VS V4 V1 Ga V3 √₂

Consider the statement, For all graphs G, if G contains an odd cycle, then x(G) > 3. Note, you do not need to know anything about graphs to answer these questions. What would the first line of a direct proof be? A. Assume there is a graph G which contains an odd cycle but for which x(G) 3. O C. Let G be a graph and assume it contains an odd cycle. O D. Let G be a graph and assume x(G) 3. O E. Let G be a graph and assume it contains no odd cycles. What would the first line of a proof by contradiction be? A. Assume there is a graph G which contains an odd cycle but for which x(G) 3.

The symmetric difference graph of two graphs G. (V. E.) and G₂ (V₁ E₂) on the same vertex Set is defined as G₁ AG₂:= (V, E, DE₂). E₁ DE ₂ = (E₁ \ E₂) U (E₂\E.) : E₁ E₂ = €₁ 0 € ₂ If G₁ and 6₂ are euleran, show that every Vertex in G, D G₂ has even degree FACT: Each vertex of a evlenan graph has even degree. SO: Show G₁ D G₂ is eulerian.