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图论代写 Graph Theory代写 Assignment代写

MT4514 Graph Theory

Assignment 1

图论代写 This assignment forms 5% of the assessment for this module. The assignment will be marked out of 20 marks.

This assignment forms 5% of the assessment for this module.

The assignment will be marked out of 20 marks.

Please answer all questions, fully justifying your answers. Think about your mathematical style – the aim is for clear, streamlined solutions. Good choice of notation can help clarify your arguments. You may cite results from the notes without proof.

From a simple graph G, we can obtain a new graph G’ as follows:

for each edge e of G, create a vertex ve of G ;

two vertices ve1 , ve2 of V (G ) are joined by an edge in G  if and only if the corresponding edges e1, e2 of E(G) share an endpoint in G.

In other words, Ghas vertices in one-to-one correspondence with the edges of G, and two vertices are adjacent in Gprecisely if the corresponding edges are adjacent in G. For example,

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(i) Let Pn denote the path on n vertices (n ≥ 2). Prove that P’n is isomorphic to Pn−1.

(ii) There exists an infinite family of graphs Gn (n 2) such that

G’n Kn−1

where Kn1 is the complete graph on n 1 vertices.

Give the vertex set and edge set of such a family, and prove that it has the stated property. (4 marks)

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Let G be a graph.

(i) Prove that if G is connected then Gis connected.

(ii) If G’ is connected, is G necessarily connected? Prove or give a counterexample. (5 marks)

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Let G be a graph. Write down a formula expressing the degree of a vertex vxy ∈ V (G’ ), in terms of the degree of x, y ∈ V (G). (2 marks)

6.

Prove that if a graph G is Eulerian, then G’ is Eulerian. (2 marks)

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Is it true that if a graph G is connected and G’ is Eulerian, then G is Eulerian? Prove or give a counterexample. (2 marks)

8.

Let G be a graph on n vertices {1, . . . , n}, where deg i = di . Prove that

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