The odd-valued chromatic polynomial of a signed graph

For a signed graph E = (G, sigma), Zaslavsky defined a proper coloring on E and showed that the function counting the number of such colorings is a quasi-polynomial with period two, that is, a pair of polynomials, one for odd values and the other for even values. In this paper, we focus on the case of odd, written as chi (E, x) for short. We initially give a homomorphism expression of such colorings, by which the symmetry is considered in counting the number of homomorphisms. Besides, the explicit formulas chi (E, x) for some basic classes of signed graphs are presented. As a main result, we give a combinatorial interpretation of the coefficients in chi(E, x) and present several applications. In particular, the constant term in chi (E, x) gives a new criterion for balancing and a characterization for unbalanced unicyclic graph. At last, we also give a tight bound for the constant term of chi(& xe002;, x).