7.02g Eulerian graphs: vertex degrees and traversability

7

1. A simple connected graph \(X\) has eight vertices.
   1. State the minimum number of edges of the graph.
   2. Find the maximum number of edges of the graph.
2. A simple connected graph \(Y\) has \(n\) vertices.
   1. State the minimum number of edges of the graph.
   2. Find the maximum number of edges of the graph.
3. A simple graph \(Z\) has six vertices and each of the vertices has the same degree \(d\).
   1. State the possible values of \(d\).
   2. If \(Z\) is connected, state the possible values of \(d\).
   3. If \(Z\) is Eulerian, state the possible values of \(d\).

7

1. The diagram shows a graph with 16 vertices and 16 edges. \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_204_365_758} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_202_365_1080} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_204_689_758} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_200_689_1082}
   1. On Figure 1 below, add the minimum number of edges to make a connected graph.
   2. On Figure 2 opposite, add the minimum number of edges to make the graph Hamiltonian.
   3. On Figure 3 opposite, add the minimum number of edges to make the graph Eulerian.
2. A complete graph has \(n\) vertices and is Eulerian.
   1. State the condition that \(n\) must satisfy.
   2. The number of edges in a Hamiltonian cycle for the graph is the same as the number of edges in an Eulerian trail. State the value of \(n\). \section*{Figure 1} \section*{Connected Graph} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{(a)(i)} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_195_200_2014_758}
      \end{figure} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{(a)(i)} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_197_200_2012_1082}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_202_2334_758}
      \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{(a)(i)} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-14_200_202_2334_1080}
      \end{figure} \section*{Figure 2} \section*{Hamiltonian Graph} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_218_536_511_753} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_213_212_836_753} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_215_214_836_1073}
      1. (iii) \section*{Eulerian Graph} \begin{figure}[h]
         \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_207_204_1356_758}
         \end{figure} \begin{figure}[h]
         \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_211_206_1354_1078}
         \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_206_209_1681_753} \includegraphics[max width=\textwidth, alt={}, center]{44bbec2c-32f4-4d28-9dd6-d89387228454-15_200_202_1681_1080}

6 The complete graph \(K _ { n } ( n > 1 )\) has every one of its \(n\) vertices connected to each of the other vertices by a single edge.

1. Draw the complete graph \(K _ { 4 }\).
2. 1. Find the total number of edges for the graph \(K _ { 8 }\).
   2. Give a reason why \(K _ { 8 }\) is not Eulerian.
3. For the graph \(K _ { n }\), state in terms of \(n\) :
   1. the total number of edges;
   2. the number of edges in a minimum spanning tree;
   3. the condition for \(K _ { n }\) to be Eulerian;
   4. the condition for the number of edges of a Hamiltonian cycle to be equal to the number of edges of an Eulerian cycle.

2

1. A graph has six vertices; two are of order 3 and the rest are of order 4. Calculate the number of arcs in the graph, showing your working.
2. Is the graph Eulerian, semi-Eulerian or neither? Give a reason to support your answer. A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is connected, directly or indirectly, to every other vertex.
3. Explain why a simple graph with six vertices, two of order 3 and the rest of order 4, must also be a connected graph.

3 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is connected, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected.

1. A simply connected graph is drawn with 6 vertices and 9 arcs.
   1. What is the sum of the orders of the vertices?
   2. Explain why if the graph has two vertices of order 5 it cannot have any vertices of order 1.
   3. Explain why the graph cannot have three vertices of order 5 .
   4. Draw an example of a simply connected graph with 6 vertices and 9 arcs in which one of the vertices has order 5 and all the orders of the vertices are odd numbers.
   5. Draw an example of a simply connected graph with 6 vertices and 9 arcs that is also Eulerian.

2

1. Draw a graph with five vertices of orders 1, 2, 2, 3 and 4 .
2. State whether the graph from part (i) is Eulerian, semi-Eulerian or neither. Explain how you know which it is.
3. Explain why a graph with five vertices of orders \(1,2,2,3\) and 4 cannot be a tree.

2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex.
A simply connected graph is one that is both simple and connected.

1. Explain why there is no simply connected graph with exactly five vertices each of which is connected to exactly three others.
2. A simply connected graph has five vertices \(A , B , C , D\) and \(E\), in which \(A\) has order \(4 , B\) has order 2, \(C\) has order 3, \(D\) has order 3 and \(E\) has order 2. Explain how you know that the graph is semi-Eulerian and write down a semi-Eulerian trail on this graph. A network is formed from the graph in part (ii) by weighting the arcs as given in the table below.

   	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
   \(A\)	-	5	3	8	2
   \(B\)	5	-	6	-	-
   \(C\)	3	6	-	7	-
   \(D\)	8	-	7	-	9
   \(E\)	2	-	-	9	-

3. Apply Prim's algorithm to the network, showing all your working, starting at vertex \(A\). Draw the resulting tree and state its total weight. A sixth vertex, \(F\), is added to the network using arcs \(C F\) and \(D F\), each of which has weight 6 .
4. Use your answer to part (iii) to write down a lower bound for the length of the minimum tour that visits every vertex of the extended network, finishing where it starts. Apply the nearest neighbour method, starting from vertex \(A\), to find an upper bound for the length of this tour. Explain why the nearest neighbour method fails if it is started from vertex \(F\).

3

1. Explain why it is impossible to draw a graph with exactly four vertices of orders 1, 2, 3 and 3 . A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex.
   A simply connected graph is one that is both simple and connected.
2. Explain why there is no simply connected graph with exactly four vertices of orders \(1,1,2\) and 4.
3. A connected graph has four vertices \(A , B , C\) and \(D\), in which \(A , B\) and \(C\) have order 2 and \(D\) has order 4. Explain how you know that the graph is Eulerian. Draw an example of such a graph and write down an Eulerian trail for your graph. A graph has three vertices, \(A , B\) and \(C\) of orders \(a , b\) and \(c\), respectively.
4. What restrictions on the values of \(a , b\) and \(c\) follow from the graph being
   1. simple,
   2. connected,
   3. semi-Eulerian?
      1. Describe carefully how to carry out the first pass through bubble sort when we are using it to sort a list of \(n\) numbers into increasing order. State which value is guaranteed to be in its correct final position after the first pass and hence explain how to carry out the second pass on a reduced list. Write down the stopping condition for bubble sort.
      2. Show the list of six values that results at the end of each pass when we use bubble sort to sort this list into increasing order. $$\begin{array} { l l l l l l } 3 & 10 & 8 & 2 & 6 & 11 \end{array}$$ You do not need to count the number of comparisons and the number of swaps that are used. Zack wants to cut lengths of wood from planks that are 20 feet long. The following lengths, in feet, are required. $$\begin{array} { l l l l l l } 3 & 10 & 8 & 2 & 6 & 11 \end{array}$$
      3. Use the first-fit method to find a way to cut the pieces.
      4. Use the first-fit decreasing method to find a way to cut the pieces. Give a reason why this might be a more useful cutting plan than that from part (iii).
      5. Find a more efficient way to cut the pieces. How many planks will Zack need with this cutting plan and how many cuts will he need to make? 5 An online shopping company selects some of its parcels to be checked before posting them. Each selected parcel must pass through three checks, which may be carried out in any order. One person must check the contents, another must check the postage and a third person must check the address. The parcels are classified according to the type of customer as 'new', 'occasional' or 'regular'. The table shows the time taken, in minutes, for each check on each type of parcel.
         1. Since \(a \leqslant 6\) it follows that \(6 - a \geqslant 0\), and similarly for \(b\) and \(c\). Let \(6 - a = x\) (so that \(a\) is replaced by \(6 - x ) , 8 - b = y\) and \(10 - c = z\) to show that the problem can be expressed as $$\begin{array} { l l } \text { Maximise } & 2 x - 4 y + 5 z , \\ \text { subject to } & 3 x + 2 y - z \leqslant 14 , \\ & 2 x - 4 z \leqslant 7 , \\ & - 4 x + y \leqslant 4 , \\ \text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$
         2. Represent the problem as an initial Simplex tableau. Perform two iterations of the Simplex algorithm, showing how each row was obtained. Hence write down the values of \(a , b\) and \(c\) after two iterations. Find the value of the objective for the original problem at this stage.
            [0pt] [10]

2

1. Draw three mathematically different graphs, labelled graph \(A\), graph \(B\) and graph \(C\), each with four vertices, of orders 1, 3, 3 and 3, and five arcs.
2. Explain how you know that none of the graphs from part (i) is Eulerian.

1 Two graphs A and B are shown below. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Write down an example of a cycle on graph A .
2. W hy is \(\mathrm { U } - \mathrm { Y } - \mathrm { V } - \mathrm { Z } - \mathrm { Y } - \mathrm { X }\) not a path on graph B ?
3. How many arcs would there be in a spanning tree for graph A ?
4. For each graph state whether it is Eulerian, semi-Eulerian or neither.
5. The graphs show designs to be etched on metal plates. The etching tool is positioned at a starting point and follows a route without repeating any arcs. It may be lifted off and positioned at a new starting point. W hat is the smallest number of times that the etching tool must be positioned, including the initial position, to draw each graph? An arc is drawn connecting Q to U , so that the two graphs become one. The resulting graph is not Eulerian.
6. Extra arcs are then added to make an Eulerian graph. W hat is the smallest number of extra arcs that need to be added?

2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex.
A simply connected graph is one that is both simple and connected.

1. Explain why it is impossible to draw a graph with exactly three vertices in which the vertex orders are 2, 3 and 4.
2. Draw a graph with exactly four vertices of orders 1, 2, 3 and 4 that is neither simple nor connected.
3. Explain why there is no simply connected graph with exactly four vertices of orders \(1,2,3\) and 4. State which of the properties 'simple' and 'connected' cannot be achieved.
4. A simply connected Eulerian graph has exactly five vertices.
   1. Explain why there cannot be exactly three vertices of order 4.
   2. By considering the vertex orders, explain why there are only four such graphs. Draw an example of each.

3 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex.
A simply connected graph is one that is both simple and connected.

1. Explain why it is impossible to draw a graph with exactly five vertices of orders \(1,2,3,4\) and 5.
2. Explain why there is no simply connected graph with exactly five vertices of orders \(2,2,3,4\) and 5 . State which of the properties 'simple' and 'connected' cannot be achieved.
3. Calculate the number of arcs in a simply connected graph with exactly five vertices of orders 1 , 1, 2, 2 and 4 . Hence explain why such a graph cannot be a tree.
4. Draw a simply connected semi-Eulerian graph with exactly five vertices that is also a tree. By considering the orders of the vertices, explain why it is impossible to draw a simply connected Eulerian graph with exactly five vertices that is also a tree. In the graph below the vertices represent buildings and the arcs represent pathways between those buildings. \includegraphics[max width=\textwidth, alt={}, center]{cec8d4db-4a72-43a3-88f3-ff9df2a11d2c-4_426_1081_1297_532}
5. By considering the orders of the vertices, explain why it is impossible to walk along these pathways in a continuous route that uses every arc once and only once. Write down the minimum number of arcs that would need to be travelled twice to walk in a continuous route that uses every arc at least once.

2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected.

1. (a) Draw a simply connected Eulerian graph with exactly five vertices and five arcs.
   (b) Draw a simply connected semi-Eulerian graph with exactly five vertices and five arcs, in which one of the vertices has order 4.
   (c) Draw a simply connected semi-Eulerian graph with exactly five vertices and five arcs, in which none of the vertices have order 4. A teacher is organising revision classes for her students. There will be ten revision classes scheduled into a number of sessions. Each class will run in one session only. Each student has chosen two classes to attend. The table shows which classes each student has chosen. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Revision classes}

   Student number	C1	C2	C3	C4	M1	M2	S1	S2	D1	D2
   1	✓	✓
   2				✓						✓
   3			✓	✓
   4			✓			✓
   5								✓	✓
   6		✓				✓
   7					✓			✓
   8	✓						✓
   9						✓				✓
   10					✓		✓

   \end{table}
2. (a) Draw a graph to show this information. Each vertex represents a class. Each arc links the two classes chosen by a student.
   (b) Show how the teacher can arrange the classes in just two sessions, which satisfy all student choices. For example, C1 and C2 cannot be in the same session. An extra student joins the group. This student chooses to attend the revision classes in M1 and D1.
   (c) Explain why the teacher cannot now arrange the classes in just two sessions. Do not amend your graph from part (ii)(a).

2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected.

1. (a) Draw a connected Eulerian graph that has exactly four vertices and five arcs but is not simple.
   (b) Explain why it is not possible to have a simply connected Eulerian graph with exactly four vertices and five arcs. A simply connected Eulerian graph is drawn that has exactly eight vertices and ten arcs.
2. (a) Explain how you know that the sum of the vertex orders must be 20 .
   (b) Write down the minimum and maximum possible vertex order and draw a diagram that includes both the minimum and the maximum cases.
   (c) Draw a diagram to show a simply connected Eulerian graph with exactly eight vertices and ten arcs in which the number of vertices of order 4 is as large as possible.

2 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected.

1. (a) Draw a simply connected graph that has exactly four vertices and exactly five arcs. Is your graph Eulerian, semi-Eulerian or neither? Explain how you know.
   (b) By considering the sum of the vertex orders, show that there is only one possible simply connected graph with exactly four vertices and exactly five arcs.
2. Draw five distinct simply connected graphs each with exactly five vertices and exactly five arcs.

4 A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected. Molly says that she has drawn a graph with exactly five vertices, having vertex orders 1, 2, 3, 4 and 5.

1. State how you know that Molly is wrong. Holly has drawn a connected graph with exactly six vertices, having vertex orders 2, 2, 2, 2, 4 and 6.
2. (a) Explain how you know that Holly's graph is not simply connected.
   (b) Determine whether Holly's graph is Eulerian, semi-Eulerian or neither, explaining how you know which of these it is. Olly has drawn a simply connected graph with exactly six vertices.
3. (a) State the minimum possible value of the sum of the vertex orders in Olly's graph.
   (b) If Olly's graph is also Eulerian, what numerical values can the vertex orders take? Polly has drawn a simply connected Eulerian graph with exactly six vertices and exactly ten arcs.
4. (a) What can you deduce about the vertex orders in Polly's graph?
   (b) Draw a graph that fits the description of Polly's graph.

1 The graph \(\mathrm { K } _ { 5 }\) has five nodes, \(A , B , C , D\) and \(E\), and there is an arc joining every node to every other node.

1. Draw the graph \(\mathrm { K } _ { 5 }\) and state how you know that it is Eulerian.
2. By listing the arcs involved, give an example of a path in \(\mathrm { K } _ { 5 }\). (Your path must include more than one arc.)
3. By listing the arcs involved, give an example of a cycle in \(\mathrm { K } _ { 5 }\).

3 Fig. 3 shows a graph representing the seven bus journeys run each day between four rural towns. Each directed arc represents a single bus journey. \begin{figure}[h] \end{figure}

1. Show that if there is only one bus, which is in service at all times, then it must start at one town and end at a different town. Give the start town and the end town.
2. Show that there is only one Hamilton cycle in the graph. Show that, if an extra journey is added from your end town to your start town, then there is still only one Hamilton cycle.
3. A tourist is staying in town B. Give a route for her to visit every town by bus, visiting each town only once and returning to B . Section B (48 marks)

1 The diagram shows the layout of a Mediterranean garden. Thick lines represent paths. \includegraphics[max width=\textwidth, alt={}, center]{aac29742-fee8-48a9-896c-e96696742251-2_961_1093_440_468}

1. Draw a graph to represent this information using the vertices listed below, and with arcs representing the 18 paths. Vertices: patio (pa); pool (po); top steps (ts); orange tree (or); fig tree (fi); pool door (pd); back door (bd); front door (fd); front steps (fs); gate (gat); olive tree (ol); garage (gar). [2] Joanna, the householder, wants to walk along all of the paths.
2. Explain why she cannot do this without repeating at least one path.
3. Write down a route for Joanna to walk along all of the paths, repeating exactly one path. Write down the path which must be repeated. Joanna has a new path constructed which links the pool directly to the top steps.
4. Describe how this affects Joanna's walk, and where she can start and finish. (You are not required to give a new route.)

1 The directed bipartite graph represents links between chairlifts and ski runs in one part of a ski resort. Chairlifts are represented by capital letters, and ski runs are represented by numbers. For example, chairlift A takes skiers to the tops of ski runs 1 and 2, whereas ski run 2 takes skiers to the bottom of chairlift B . \includegraphics[max width=\textwidth, alt={}, center]{a27c868b-4fc4-4e82-b27f-d367b15b42c2-2_551_333_493_849}

1. The incomplete map in your answer book represents the three chairlifts and ski run 2 . Complete the map by drawing in the other 4 ski runs. Angus wants to ski all 5 ski runs, starting and finishing at the bottom of chairlift A .
2. Which chairlifts does Angus have to repeat, and why?
3. Which ski runs does Angus have to repeat, and why? The chairlifts and ski runs shown above form only part of the resort. In fact, chairlift C also takes skiers to the bottom of chairlift \(D\).
4. Why can this information not be represented in a bipartite graph?

2 A talent contest has five contestants. In the first round of the contest each contestant must sing a song chosen from a list. No two contestants may sing the same song. Adam (A) chooses to sing either song 1 or song 2; Bex (B) chooses 2 or 4; Chris (C) chooses 3 or 5; Denny (D) chooses 1 or 3; Emma (E) chooses 3 or 4.

1. Draw a bipartite graph to show this information. Put the contestants (A, B, C, D and E) on the left hand side and the songs ( \(1,2,3,4\) and 5 ) on the right hand side. The contest organisers propose to give Adam song 1, Bex song 2 and Chris song 3.
2. Explain why this would not be a satisfactory way to allocate the songs.
3. Construct the shortest possible alternating path that starts from song 5 and brings Denny (D) into the allocation. Hence write down an allocation in which each of the five contestants is given a song that they chose.
4. Find a different allocation in which each of the five contestants is given a song that they chose. Emma is knocked out of the contest after the first round. In the second round the four remaining contestants have to act in a short play. They will each act a different character in the play, chosen from a list of five characters. The table below shows how suitable each contestant is for each character as a score out of 10 (where 0 means that the contestant is completely unsuitable and 10 means that they are perfect to play that character).

   \multirow{2}{*}{}	Character
   	Fire Chief	Gardener	Handyman	Juggler	King
   Adam	4	9	7	0	7
   Bex	6	8	3	8	0
   Chris	7	4	5	2	7
   Denny	6	6	2	7	1

   The Hungarian Algorithm is to be used to find the matching with the greatest total score. Before the Hungarian Algorithm can be used, each score is subtracted from 10 and then a dummy row of zeroes is added at the bottom of the table.
5. Explain why the scores could not be used as given in the table and explain why a dummy row is needed.
6. Apply the Hungarian Algorithm, showing your working carefully, to match the contestants to characters.

1

1. A café sells five different types of filled roll. Mr King buys one of each to take home to his family. The family consists of Mr King's daughter Fiona ( \(F\) ), his mother Gwen ( \(G\) ), his wife Helen ( \(H\) ), his son Jack ( \(J\) ) and Mr King ( \(K\) ). The table shows who likes which rolls.

   		\(F\)	\(G\)	\(H\)	\(J\)	\(K\)
   Avocado and bacon	\(( A )\)	\(\checkmark\)		\(\checkmark\)
   Beef and horseradish	\(( B )\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
   Chicken and stuffing	\(( C )\)		\(\checkmark\)		\(\checkmark\)
   Duck and plum sauce	\(( D )\)			\(\checkmark\)		\(\checkmark\)
   Egg and tomato	\(( E )\)	\(\checkmark\)

   1. Draw a bipartite graph to represent this information. Put the fillings ( \(A , B , C , D\) and \(E\) ) on the left-hand side and the members of the family ( \(F , G , H , J\) and \(K\) ) on the right-hand side. Fiona takes the avocado roll; Gwen takes the beef roll; Helen takes the duck roll and Jack takes the chicken roll.
   2. Draw a second bipartite graph to show this incomplete matching.
   3. Construct the shortest possible alternating path from \(E\) to \(K\) and hence find a complete matching. State which roll each family member has with this complete matching.
   4. Find a different complete matching.
2. Mr King decides that the family should eat more fruit. Each family member gives a score out of 10 to five fruits. These scores are subtracted from 10 to give the values below. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Family member}

   		\(F\)	\(G\)	\(H\)	\(J\)	\(K\)
   Lemon	\(L\)	8	8	8	8	1
   Mandarin	\(M\)	4	8	6	4	2
   Nectarine	\(N\)	9	9	9	7	1
   Orange	\(O\)	4	6	5	4	3
   Peach	\(P\)	6	9	7	5	0

   \end{table} The smaller entries in each column correspond to fruits that the family members liked most.
   Mr King buys one of each of these five fruits. Each family member is to be given a fruit.
   Apply the Hungarian algorithm, reducing rows first, to find a minimum cost matching. You must show your working clearly. Which family member should be given which fruit?

1 Adam, Barbara and their children Charlie, Donna, Edward and Fiona all want cereal for breakfast. The only cereal in the house is a pack of six individual portions of different cereals. The table shows which family members like each of the cereals in the pack.

1. Draw a bipartite graph to represent this information. Adam gives the cornflakes to Fiona, the oatie bits to Edward, the rice pips to Donna, the choco pips to Charlie and the wheat biscs to Barbara. However, this leaves the honey footballs for Adam, 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 6 to \(A\) and hence find a complete matching between the cereals and the family members. Write down which family member is given each cereal with this complete matching.
4. Adam decides that he wants cornflakes. Construct an alternating path starting at \(A\), based on your answer to part (iii) but with Adam being matched to the cornflakes, to find another complete matching. Write down which family member is given each cereal with this matching.

1 The six cadets in Red Team have been told to guard a building through the night, starting at 2200 hours and finishing at 0800 hours the next day. Each will be on duty for either one hour or three hours and will then hand over to the next cadet. The table shows which duty each cadet has offered to take. \begin{table}[h] \end{table}

1. Draw a bipartite graph to represent this information. Amir suggests that he should take the 2200 duty, hand over to Becca at 0100 , she can hand over to Chris at 0200 , and Dan can take the 0400 duty. However, this leaves Emma and Finn to cover the 0300 and 0500 duties, and neither of them wants either of these.
2. Write down the shortest possible alternating path starting at the 0500 duty and hence write down an improved but still incomplete matching between the cadets and the duties.
3. Augment this second incomplete matching by writing down a shortest possible alternating path, this time starting from one of the cadets, to form a complete matching between the cadets and the duties. Write down which cadet should take which duty.

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

1. List the vertex degrees for each graph.
2. Prove that the graphs are non-isomorphic. The two graphs are joined together by adding an arc connecting J and T .
3. 1. Explain how you know that the resulting graph is not Eulerian.
   2. Describe how the graph can be made Eulerian by adding one more arc. The vertices of the graph \(K _ { 3 }\) are connected to the vertices of the graph \(K _ { 4 }\) to form the graph \(K _ { 7 }\).
4. Explain why 12 arcs are needed connecting \(K _ { 3 }\) to \(K _ { 4 }\).