Generalized fractional total colorings of graphs

Let P and Q be additive and hereditary graph properties and let r, s be integers such that r ≥ s. Then an r s -fractional (P,Q)-total coloring of a finite graph G = (V,E) is a mapping f , which assigns an s-element subset of the set {1, 2, . . . , r} to each vertex and each edge, moreover, for any color i all vertices of color i induce a subgraph with property P, all edges of color i induce a subgraph with property Q and vertices and incident edges have been assigned disjoint sets of colors. The minimum ratio of an r s -fractional (P,Q)-total coloring of G is called fractional (P,Q)-total chromatic number χ f,P,Q(G) = r s . We show in this paper that χ f,P,Q of a graph G with o(V (G)) vertex orbits and o(E(G)) edge orbits can be found as a solution of a linear program with integer coefficients which consists only of o(V (G)) + o(E(G)) inequalities.