Use proof by contradiction to show that a cycle graph with an odd number of nodes...

Use proof by contradiction to show that a cycle graph with an odd number of nodes is not 2-colorable.

Solutions

Expert Solution

Assumption-

Suppose, if possible a cycle graph with an odd number of nodes is 2-colorable. Then this must be true for any cycle graph with an odd number of nodes.

Construction-

Suppose, we take a cycle graph with nodes 3 (an odd number).

Thus our graph is as follows.

Graph coloring-

Suppose, we start coloring from node 1. We use any color x1 (say yellow) in this node.
We proceed to node 2 and color it. As color x1 is already used in neighboring node 1, it cannot be used in node 2. As the graph is 2-colorable (by assumption), we can take another color x2 (say blue). We color node 2 with second color x2.
We now proceed to next node i.e. node 3. Node 3 is adjacent to nodes 1 and node 2. So, none of the colors used in nodes 1 and 2 can be used in coloring node 3. So, we have to take any color other than x1 and x2. But it was assumed that the graph is 2-colorable. So, we cannot take a 3rd color into consideration. This arises the situation that the graph coloring is incomplete with 2 colors.

Hence, by the method of contradiction we can conclude that a cycle graph with an odd number of nodes is not 2-colorable.

orchestra answered 1 year ago

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