Embedding equitable (s,p)-edge-colorings of $K_n$

Source abstract

An $(s,p)$-edge-coloring of a graph $G$ is an edge coloring using $s$ colors such that exactly $p$ colors appear at each vertex. To generalize the notion of proper edge-coloring, these colorings are defined to be equitable: the numbers of edges of each color incident to a vertex are fairly distributed. We find the necessary and sufficient conditions for embedding an equitable $(s_1,p_1)$-edge-coloring of $K_{n_1}$ into an equitable $(s_2,p_2)$-edge-coloring of $K_{n_2}$. We focus on the values of $n_1$, $p_1$, $n_2$, and $p_2$ where $s_1$ is necessarily larger than $p_1$ and $s_2$ is necessarily larger than $p_2$.