On coloring the arcs of biregular graphs

Abstract

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 (λ,μ)-biregular graph with bipartition (Y,X), where |Y|=kμ and |X|=kλ, (0<k,λ,μ∈Z). We consider the problem, for each edge e=yx in H, of assigning, a color (given by an element) of Y, resp. X, to the arc (y,e), resp. (x,e), so that each 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 Y×X. This problem applies to the Design of Experiments for Industrial Chemistry, Molecular Biology, Cellular Neuroscience, etc. An algorithmic construction based on biregular graphs with bipartitions given by cyclic-group pairs is presented, as well as 3 essentially different solutions to the Great Circle Challenge Puzzle based on a different biregular graph whose bipartition is formed by the vertices and 5-cycles of the Petersen graph.