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look  if a graph is not complete then chromatic no is dmax.means higest degree of that graph.

u can check it for k4 and others.

so if we remove an edge from kn.then chromatic no is n-1.
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n-1

where n is nodes.

in complete every1 is adjacent to each other. so we require n colours.

but removal of any edge causes one less adjacency.so N-1

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