Reference no: EM131238904
1. Consider the following labelled graph
(a) State an upper-bound and a lower-bound on the vertex-chromatic number of the graph, and give reasons for your answers.
(b) Find an optimal colouring of the graph.
(c) To which of the following classes does the graph belong: chordal graphs, trees, bipartite graphs, perfect graphs? Provide reasons for your answers.
(d) Beginning with the matching M = {(e, f )}, extend M into a maximum matching by repeatedly finding augmenting paths. At each step you should increase the number of edges in the matching by one. Explicitly show each step of the process.
Recall: an augmenting path is an alternating path that joins two "exposed" vertices. An edge between two exposed nodes is also regarded as an augmenting path.
(e) State a lower-bound on the edge-chromatic number of the graph, and give a reason for your answer.
2. Suppose teachers x1, x2, x3, x4 have to teach classes y1, y2, y3, y4, y5, y6 according to the following teacher/class allocation table:
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y1
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y2
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y3
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y4
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y5
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y6
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x1
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0
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0
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1
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0
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1
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1
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x2
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1
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0
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1
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0
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0
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2
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x3
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1
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1
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1
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0
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0
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1
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x4
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2
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0
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0
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1
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1
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0
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(a) What is the minimum possible number of periods used in this allocation. How do we know this without doing a decomposition?
(b) Calculate the minimum number of periods using a single 1's decomposition.
(c) Construct a timetable using the minimum number of periods.
3. For n Ç 3, find the number of distinct perfect matchings in the cycle Cn of length n. Justify your answer.
4. Suppose that T is a tree (a connected graph without cycles) with n vertices and that every vertex of T has degree 1 or 3.
(a) Prove rigorously that n is even.
(b) Prove rigorously that T has exactly (n + 2)/2 leaves (vertices of degree 1).
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Synthesize appropriate current research
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Determine an expression for the fundamental axial mode shape
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Plywood firm uses three inputs-labor-lumber and saws
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Find an optimal colouring of the graph
: MAST20018 - Discrete Mathematics and Operations Research State an upper-bound and a lower-bound on the vertex-chromatic number of the graph, and give reasons for your answers - Find an optimal colouring of the graph.
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What is concept of family most used in your area of nursing
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