$K_{s,t}$-Polychromatic edge-colorings of complete bipartite graphs

Abstract:

      Let $G$ be a graph and $\mathcal{H}$ be a set of subgraphs of $G$. An $h$-edge-coloring of $G$ is $\mathcal{H}$-polychromatic if every subgraph of $G$ isomorphic to some element in $\mathcal{H}$ receives all $h$ colors. The largest integer $h$, for which $G$ admits an $\mathcal{H}$-polychromatic $h$-edge-coloring, is called the $\mathcal{H}$-polychromatic number of $G$ and denoted by $p_{\mathcal{H}}(G)$. In this paper, we prove that $p_{K_{s,t}}(K_{m,n})=\lfloor\frac{mn}{m+n-s-t+1}\rfloor$ for $2\leq s