On r-equitable chromatic threshold of Kronecker products of complete graphs

A graph G is r-equitably k-colorable if its vertex set can be partitioned into k independent sets, any two of which differ in size by at most r. The r-equitable chromatic threshold of a graph G, denoted by χr=(G), is the minimum k such that G is r-equitably k -colorable for all k ≥ k. Let G × H denote the Kronecker product of graphs G and H. In this paper, we completely determine the exact value of χr=(Km × Kn) for general m, n and r. As a consequence, we show that for r ≥ 2, if n ≥ 1 r−1 (m+ r)(m+2r −1) then Km ×Kn and its spanning supergraph Km(n) have the same r-equitable colorability, and in particular χr=(Km × Kn) = χ ∗ r=(Km(n)), where Km(n) is the complete m-partite graph with n vertices in each part.