13. Suppose you colored edges of a graph either red or blue (not requiring that, adjacent edges be colored differently). What must be true of the graph to guarantee some vertex is incident to three edges of the same color? Prove your answer. y Hint • Solution ANA....u..... You can color K5 in such a way that every vertex is adjacent to exactly two blue edges and two red edges. However, there is a graph with only 5 edges that will result in a vertex incident to three edges of the same color no matter how they are colored. What is it, and how can you generalize? 14. Prove that if you color every edge of K, either red or blue. you are guaranteed a monochromatic triangle (that is, an all red or an all blue triangle). * Hint The previous exercise is useful as a starting polnt.
13. Suppose you colored edges of a graph either red or blue (not requiring that, adjacent edges be colored differently). What must be true of the graph to guarantee some vertex is incident to three edges of the same color? Prove your answer. y Hint • Solution ANA....u..... You can color K5 in such a way that every vertex is adjacent to exactly two blue edges and two red edges. However, there is a graph with only 5 edges that will result in a vertex incident to three edges of the same color no matter how they are colored. What is it, and how can you generalize? 14. Prove that if you color every edge of K, either red or blue. you are guaranteed a monochromatic triangle (that is, an all red or an all blue triangle). * Hint The previous exercise is useful as a starting polnt.
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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Number 14 please.
Transcribed Image Text:13. Suppose you colored edges of a graph either red or blue (not requiring that,
adjacent edges be colored differently). What must be true of the graph to
guarantee some vertex is incident to three edges of the same color? Prove your
answer.
y Hint
• Solution
ANA....u.....
You can color K5 in such a way that every vertex is adjacent to exactly two
blue edges and two red edges. However, there is a graph with only 5 edges
that will result in a vertex incident to three edges of the same color no matter
how they are colored. What is it, and how can you generalize?
14. Prove that if you color every edge of K, either red or blue. you are
guaranteed a monochromatic triangle (that is, an all red or an all blue triangle).
* Hint
The previous exercise is useful as a starting polnt.
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