Discrete Mathematics with Graph Theory (3rd Edition) 380

Discrete - 378 9 Here X(Q = 4 To see this start with vertex V l and proceed clockwise Since V l V 2 V 3 f onn a triangle they must b e given

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
378 9. Here X(Q) = 4. To see this, start with vertex Vl and proceed clockwise. Since Vl, V2, V3 fonn a triangle, they must be given different colors, say red, white and green. To avoid the use of a fourth colour, we would have to color V4 red (since it has already been joined to green and white vertices). But then V5 must be white and V6 green. Now V7 is adjacent to red, white and green vertices, so we need a fourth colour, blue, for this. 10. For the graph on the left, X(Q) = 3. Since the graph contains triangles, X(Q) ~ 3, and a 3-coloring is shown. For the graph on the right, X(Q) = 4. Since the graph contains a copy of lC4, X(Q) ~ 4, and a 4-coloring is shown. Each graph reminds us that the converse to the Four-Color Theorem is false since in Exercise 2 we noted that neither is planar, yet X(9) :::; 4 in each case. 11. (a) False. A counterexample is shown. ~ Solutions to Review Exercises (b) True. Since 9 contains lC4 as a subgraph, we know that X(g) ~ 4. But since 9 is planar, X(g)
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

Ask a homework question - tutors are online