I can explain how I got 5 but is it right? Otherwise I should try something else.

$\\ Draw\ a\ K_{4,5}. \\ Now\ when\ you\ will\ complement\ it\ you\ will\ get\ K_{4}\ and\ K_{5}\\ Means \ a\ disconnected\ graph\ with\ two\ components\\ For\ K_{5} \ you\ need\ 5 \ colors\ and\ out \ of\ those\ pick\ any \ 4 \ colors \ and \ color\ K_{4}.\\ So\ chromatic\ number=5$