  In this paper, we present a minimal chordal completion $G^*$ of a graph $G$
satisfying the inequality $\omega(G^*) - \omega(G) \le i(G)$ for the
non-chordality index $i(G)$ of $G$. In terms of our chordal completions, we
partially settle the Hadwiger conjecture and the Erd\H{o}s-Faber-Lov\'{a}sz
Conjecture, and extend the known $\chi$-bounded class by adding to it the
family of graphs with bounded non-chordality indices.
