Rainbow domination numbers on graphs with given radius

For k≥1, r≥1 and n≥1, let t∗(k,r;n) be the minimum value satisfying that γrk(G)≤t∗(k,r;n)⋅n for any connected graph G of order n with radius r; if no such graph exists, we set t∗(k,r;n)=∞. For k≥1 and r≥1, let t∗(k,r)=lim supn→∞t∗(k,r;n). In this paper, we investigate the behavior of the function t∗(k,r) and determine some exact values of t∗(k,r) when k or r is small.