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 chi(f,P,Q)''(G) = r/s. We show in this paper that chi(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.