Combinatorics / CONCEPT

Reciprocal estimates for singleton color leaves

Let G be a simple graph on a finite vertex type V, and let c:V->Fin(3) be a proper coloring: adjacent vertices have different colors. Write A_i={x:c(x)=i} and B=sum_i 1/(card(A_i)+1)+(1/2)sum_x 1/(degree_G(x)+1), with rational arithmetic. All three colors are included in the first sum, and every vertex, including each isolated vertex, is included in the second sum.