An extension of the bivariate chromatic polynomial

K.Dohmen, A.Pönitz and P.Tittman (2003), introduced a bivariate generalization of the chromatic polynomial P (G, x, y) which subsumes also the independent set polynomial of I. Gutman and F. Harary, (1983) and the vertex-cover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little (2002). We first show that P (G, x, y) has a recursive definition with respect to three kinds of edge elimination: edge deletion, edge contraction, and and edge extraction, i.e. deletion of an edge together with its end points. Like in the case of deletion and contraction only (J.G. Oxley and D.J.A. Welsh 1979) it turns out that there is a most general, or as they call it, a universal polynomial satisfying such a recurrence relations with respect to the three kinds of edge elimations, which we call ξ(G, x, y, z). We show that the new polynomial simultaneously generalizes, P (G, x, y), as well as the Tutte polynomial and the matching polynomial, We also give an explicit definition of ξ(G,x, y, z) using a subset expansion formula. We also show that ξ(G,x, y, z) can be viewed as a partition function, using counting of weighted graph homomorphisms. Furthermore, we expand this result to edge-labeled graphs as was done for the Tutte polynomial by T. Zaslavsky (1992) and by B. Bollobas and O. Riordan (1999). The edge labeled polynomial ξlab(G, x, y, z, t̄) also generalizes the chain polynomial of R.C. Read and E.G. Whitehead Jr. (1999). Finally, we discuss the complexity of computing ξ(G, x, y, z).