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4. All graphs in this question are finite and simple. (a) If T is a tree with at least 2 vertices, show that X(T) = 2. (b) Determine x(Cn, k), the number of k colourings of the cycle Cn with n vertices, for n> 3. In parts (c) and (d) below, G is a graph such that x(G) > x(G – v) for all v € V(G). That is, for any v E V(G) the graph G-v has a colouring with less colours than G does. (c) Show that G is connected. (d) Show that dG(v) > x(G) - 1 for all v € V(G), that is, all vertices of G have degree at least X(G) - 1.

4. All graphs in this question are finite and simple. (a) If T is a tree with at least 2 vertices, show that X(T) = 2. (b) Determine x(Cn, k), the number of k colourings of the cycle Cn with n vertices, for n> 3. In parts (c) and (d) below, G is a graph such that x(G) > x(G – v) for all v € V(G). That is, for any v E V(G) the graph G-v has a colouring with less colours than G does. (c) Show that G is connected. (d) Show that dG(v) > x(G) - 1 for all v € V(G), that is, all vertices of G have degree at least X(G) - 1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Discreet math, combinatorics and graphs

4. All graphs in this question are finite and simple. (a) If T is a tree with at least 2 vertices, show that x(T) = 2. (b) Determine x(Cn, k), the number of k colourings of the cycle Cn with n vertices, for n> 3. - In parts (c) and (d) below, G is a graph such that x(G) > x(G – v) for all v = V(G). That is, for any v € V (G) the graph G – v has a colouring with less colours than G does. (c) Show that G is connected. (d) Show that da(v) ≥ x(G) – 1 for all v € V(G), that is, all vertices of G have degree at least x(G) - 1.
Transcribed Image Text:4. All graphs in this question are finite and simple. (a) If T is a tree with at least 2 vertices, show that x(T) = 2. (b) Determine x(Cn, k), the number of k colourings of the cycle Cn with n vertices, for n> 3. - In parts (c) and (d) below, G is a graph such that x(G) > x(G – v) for all v = V(G). That is, for any v € V (G) the graph G – v has a colouring with less colours than G does. (c) Show that G is connected. (d) Show that da(v) ≥ x(G) – 1 for all v € V(G), that is, all vertices of G have degree at least x(G) - 1.
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