On coloring the arcs of biregular graphs

Recalling each edge of a graph $H$ has 2 oppositely oriented arcs, each vertex $v$ of $H$ is identified with the set of arcs, denoted $(v,e)$, departing from $v$ along the edges $e$ of $H$ incident to $v$. Let $H$ be a $(\lambda,\mu)$-biregular graph with vertex parts $Y$ and $X$, where $|Y|$, resp. $|X|$, is a multiple of the positive integer $\mu$, resp. $\lambda$ ($\ne\mu$). We consider the problem, for each edge $e=yx$ in $H$, of assigning, a {\it color} (given by an element) of $\Z_\lambda$, resp. $\Z_\mu$, to the arc $(y,e)$, resp. $(x,e)$, so that each such color is assigned exactly once in the set of arcs departing from each vertex of $H$. Furthermore, we set such assignment to fulfill a specific bicolor weight function over a monotonic subset of $\Z_\lambda\times \Z_\mu$. This problem applies to the Design of Experiments for Industrial Chemistry, Molecular Biology, Cellular Neuroscience, etc. Four constructions of such arc colorings are presented: an algorithmic one based on pairs of cyclic groups, and 3 essentially different ones based on the vertices and 5-cycles of the Petersen graph. Each of these 3 constructions applies to assembling the Great Circle Challenge Puzzle.