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.

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter1: Equations And Graphs

Section1.2: Graphs Of Equations In Two Variables; Circles

Problem 5E: a If a graph is symmetric with respect to the x-axis and (a,b) is on the graph, then (,) is also on...

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