A new minimal chordal completion

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.