The goal of this problem is to show that a graph is 2-colorable if and only if it contains no closed walks of odd length

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The Goal Of This Problem Is To Show That A Graph Is 2 Colorable If And Only If It Contains No Closed Walks Of Odd Length 1 (98.75 KiB) Viewed 40 times

The goal of this problem is to show that a graph is 2-colorable if and only if it contains no closed walks of odd length. a) Prove that if a graph contains a closed walk of odd length then it is not 2-colorable. b) Prove that every tree is 2-colorable. c) Prove that every forest is 2-colorable. d) Suppose that G contains contains no closed walks of odd length. Let H be a maximal spanning forest of G and let f be a 2-coloring of H. Prove that f is a 2-coloring of G, by showing that every edge of G that isn't in H must connect two differently-colored vertices.