7.02i Ore's theorem: sufficient condition for Hamiltonian graphs

4 A connected graph is shown below. \includegraphics[max width=\textwidth, alt={}, center]{50697293-6cdc-475f-981f-71a351b0ff5a-4_442_954_296_246}

1. Write down a path through exactly 7 of the vertices.
2. Write down a cycle through exactly 6 of the vertices.
3. Explain why Ore's theorem cannot be used to decide whether or not this graph is Hamiltonian.
4. Prove that the graph is not Hamiltonian. The following colouring algorithm can be used to determine whether a connected graph is bipartite or not. The algorithm colours each vertex of a graph in one of two colours, (1) and (2). STEP 1 Choose a vertex and assign it colour (1).
   STEP 2 If any vertex is adjacent to another vertex of the same colour, stop. Otherwise assign colour (2) to each vertex that is adjacent to a vertex with colour (1).
   STEP 3 If any vertex is adjacent to another vertex of the same colour, stop. Otherwise assign colour (1) to each vertex that is adjacent to a vertex with colour (2).
   STEP 4 Repeat STEP 2 and STEP 3 until all vertices are coloured.
   STEP 5 If there are no adjacent vertices of the same colour then the graph is bipartite, output the word "bipartite".
   Otherwise the graph is not bipartite, output the words "not bipartite".
5. Use this algorithm, starting at vertex A, to determine whether the graph is bipartite, or not. [2
6. Explain what Kuratowski's theorem tells you about the graph.
7. Show that the graph has thickness 2 .

1 This question is about the planar graph shown below. \includegraphics[max width=\textwidth, alt={}, center]{cc58fb7a-efb6-4548-a8e1-e40abe1eb722-2_567_1317_395_374}

1. 1. Apply Kuratowski's theorem to verify that the graph is planar.
   2. Use Euler's formula to calculate the number of regions in a planar representation of the graph.
2. 1. Write down a Hamiltonian cycle for the graph.
   2. By finding a suitable pair of vertices, show that Ore's theorem cannot be used to prove that the graph, as shown above, is Hamiltonian.
3. 1. Draw the graph formed by using the contractions AB and CF .
   2. Use Ore's theorem to show that this contracted graph is Hamiltonian.

4. (a) Explain why it is not possible to draw a graph with exactly six nodes with degrees 1, 2, 3, 4, 5 and 6 A tree, T , has exactly six nodes. The degrees of the six nodes of T are
1
2 \(( 4 - x )\) \(( 2 x - 5 )\) \(( 4 x - 11 )\) \(( 3 x - 5 )\) where \(x\) is an integer.
(b) Explain how you know that T cannot be Eulerian.
(c) (i) Determine the value of \(x\) (ii) Hence state whether T is semi-Eulerian or not. You must justify your answer.
(5) \includegraphics[max width=\textwidth, alt={}, center]{7f7546eb-0c1a-40da-bdf0-31e0574a9867-07_588_579_977_744} \section*{Figure 2} Figure 2 shows a graph, \(G\), with six nodes with degrees \(1,2,3,3,3\) and 4
(d) Using the vertices in Diagram 1 in the answer book, draw a graph with exactly six nodes with degrees \(1,2,3,3,3\) and 4 that is not isomorphic to G .

2 Two simply connected graphs are shown below. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. 1. Write down the orders of the vertices for each of these graphs.
   2. How many ways are there to allocate these vertex degrees to a graph with vertices \(\mathrm { P } , \mathrm { Q }\), \(\mathrm { R } , \mathrm { S } , \mathrm { T }\) and U ?
   3. Use the vertex degrees to deduce whether the graphs are Eulerian, semi-Eulerian or neither.
2. Show that graphs 1 and 2 are not isomorphic.
3. 1. Write down a Hamiltonian cycle for graph 1.
   2. Use Euler's formula to determine the number of regions for graph 1.
   3. Identify each of these regions for graph 1 by listing the cycle that forms its boundary.

1 The famous fictional detective Agatha Parrot is investigating a murder. She has identified six suspects: Mrs Lemon \(( L )\), Prof Mulberry \(( M )\), Mr Nutmeg \(( N )\), Miss Olive \(( O )\), Capt Peach \(( P )\) and Rev Quince \(( Q )\). The table shows the weapons that could have been used by each suspect.

1. Draw a bipartite graph to represent this information. Put the weapons on the left-hand side and the suspects on the right-hand side. Agatha Parrot is convinced that all six suspects were involved, and that each used a different weapon. She originally thinks that the axe handle was used by Prof Mulberry, the broomstick by Miss Olive, the drainpipe by Mrs Lemon, the fence post by Mr Nutmeg and the golf club by Capt Peach. However, this would leave the hammer for Rev Quince, which is not a possible pairing.
2. Draw a second bipartite graph to show this incomplete matching.
3. Construct the shortest possible alternating path from \(H\) to \(Q\) and hence find a complete matching. Write down which suspect used each weapon.
4. Find a different complete matching in which none of the suspects used the same weapon as in the matching from part (iii).

4 The connected planar graph \(P\) has the adjacency matrix 4

1. Draw \(P\) 4
2. Using Euler's formula for connected planar graphs, show that \(P\) has exactly 5 faces. 4
3. Ore's theorem states that a simple graph with \(n\) vertices is Hamiltonian if, for every pair of vertices \(X\) and \(Y\) which are not adjacent, $$\text { degree of } X + \text { degree of } Y \geq n$$ where \(n \geq 3\) Using Ore's theorem, prove that the graph \(P\) is Hamiltonian.
   Fully justify your answer.

8 The graph \(G\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-12_301_688_351_676} 8

1. 1. State, with a reason, whether or not \(G\) is simple. 8
      1. (ii) A student states that \(G\) is Eulerian.
         Explain why the student is correct. 8
   2. The graph \(H\) has 8 vertices with degrees 2, 2, 4, 4, 4, 4, 4 and 4 Comment on whether \(H\) is isomorphic to \(G\) 8
   3. The formula \(v - e + f = 2\), where \(v =\) number of vertices \(e =\) number of edges \(f =\) number of faces
      can be used with graphs which satisfy certain conditions. Prove that \(G\) does not satisfy the conditions for the above formula to apply.

The graph below connects nine vertices A, B, \(\ldots\), H, I. \includegraphics{figure_2}

1. 1. Show that the minimum sum of the degrees of each pair of non-adjacent vertices is 9. [2]
   2. Explain what you can deduce from the result in part (a). [1]
2. Use Kuratowski's theorem to prove that the graph is non-planar. [3]
3. Prove that there is no subgraph of the graph that is isomorphic to \(K_4\), without using subdivision or contraction. [6]