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.