Homework Set # 4 — Solution Notes
#1, 2.3: (j). The graph is bipartite, only 2 colours are needed.
= 4. There is a wheel formed by nodes a,b,d,j,i,c, as in Example 2 so we need at least 4 colours.
Since the graph is planar, 4 colours suFce, for example: a-2, b-4, c-3, d-1, e-3, f-1, g-2, h-4, i-1, j-2.
#2, 2.3: (b). 3 colours are needed since nodes have degree 3. 3 colours are enough: (
) and (
colour 1, (
) and (
) colour 2, and the remaining 3 edges colour 3.
(d) By Vizing’s Theorem, we know we ned either 4 or 5 colours, since the max node degree is 4. In fact
4 colours are enough: Colour 1 are edges (e,a) (b,c) and (d,g). Colour 2 are edges (a,b) (e,g) (c,f). Colour
3 are edges (a,d) (b,e) (f,g). Colour 4 are edges (a,g) (c,d) (e,f). Note: the 4 colour Theorem is not helpful
here, as it is about node colouring, not edge colouring.