On a relation between the Randić index and the chromatic number

The Randić index of a graph GG, denoted by R(G)R(G), is defined as the sum of 1/d(u)d(v) over all edges uvuv of GG, where d(u)d(u) denotes the degree of a vertex uu in GG. Caporossi and Hansen proposed a conjecture on the relation between the Randić index R(G)R(G) and the chromatic number χ(G)χ(G) of a graph GG: for any connected graph GG of order n≥2n≥2, R(G)≥χ(G)−22+1n−1(χ(G)−1+n−χ(G)), and furthermore the bound is sharp for all nn and 2≤χ(G)≤n2≤χ(G)≤n. We prove this conjecture.