On a relation between the Randić index and the chromatic number

The Randić index of a graph G , denoted by R ( G ) , is defined as the sum of 1 / d ( u ) d ( v ) over all edges u v of G , where d ( u ) denotes the degree of a vertex u in G . Caporossi and Hansen proposed a conjecture on the relation between the Randić index R ( G ) and the chromatic number χ ( G ) of a graph G : for any connected graph G of order n ≥ 2 , R ( G ) ≥ χ ( G ) − 2 2 + 1 n − 1 ( χ ( G ) − 1 + n − χ ( G ) ) , and furthermore the bound is sharp for all n and 2 ≤ χ ( G ) ≤ n . We prove this conjecture.