The multicolorings of graphs and hypergraphs

Abstract

Let H=(X, ξ) be a hypergraph with vertex-set X={x₁, x₂, ..., x_n}. We have a multicoloring of H with λ colors if we can assign to each X ε X one or several of these colors so that in each edge E ε ξ every color occurs exactly once.

Clearly, if there exists a multicoloring of H, one color defines a set of vertices which is both strongly stable and transversal; no general condition for the existence of such a set exists in the literature. The purpose of this paper is to provide existence conditions not only for a strongly stable transversal set, but also for a multicoloring.