Statement. Let G be a connected graph. G is bipartite if and only if x(G) = 2. (a) We will first prove P→Q. So we are assuming G is a connected, bipartite graph with partite sets X and Y, and we'll prove x(G) = 2. i. How do you color the vertices of the graph using only the colors 1 and 2 so that you have a proper vertex-coloring? Provide some justification as to why this will be a proper vertex- coloring. (This will show that x(G) ≤ 2). ii. Why is x(G) 2? (b) We now prove Q→ P. So we are assuming G is a connected graph with x(G) = 2, and we'll prove that G is bipartite (you do NOT know G is bipartite!). i. In order to show G is bipartite, we must identify partite sets X and Y. What are they? Why is there no edge that has both endpoints in X or both endpoints in Y?
Statement. Let G be a connected graph. G is bipartite if and only if x(G) = 2. (a) We will first prove P→Q. So we are assuming G is a connected, bipartite graph with partite sets X and Y, and we'll prove x(G) = 2. i. How do you color the vertices of the graph using only the colors 1 and 2 so that you have a proper vertex-coloring? Provide some justification as to why this will be a proper vertex- coloring. (This will show that x(G) ≤ 2). ii. Why is x(G) 2? (b) We now prove Q→ P. So we are assuming G is a connected graph with x(G) = 2, and we'll prove that G is bipartite (you do NOT know G is bipartite!). i. In order to show G is bipartite, we must identify partite sets X and Y. What are they? Why is there no edge that has both endpoints in X or both endpoints in Y?
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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Transcribed Image Text:Statement. Let G be a connected graph. G is bipartite if and only if x(G) = 2.
(a) We will first prove P→Q. So we are assuming G is a connected, bipartite graph with partite
sets X and Y, and we'll prove x(G) = 2.
i. How do you color the vertices of the graph using only the colors 1 and 2 so that you have
a proper vertex-coloring? Provide some justification as to why this will be a proper vertex-
coloring. (This will show that x(G) ≤ 2).
ii. Why is x(G) 2?
(b) We now prove Q→ P. So we are assuming G is a connected graph with x(G) = 2, and we'll
prove that G is bipartite (you do NOT know G is bipartite!).
i. In order to show G is bipartite, we must identify partite sets X and Y. What are they? Why
is there no edge that has both endpoints in X or both endpoints in Y?
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