Chromatic sums of singular maps on some surfaces. (Q1880478)

A singular map is a graph drawn crossing-free on a surface such that each edge is on the same face of that surface. So singular maps on the plane or sphere are just trees, whereas for other surfaces more complicated graphs are possible. The authors consider the families of rooted singular maps on the projective plane, torus, and Klein bottle. They define the chromatic sum function of a family of rooted graphs as a polynomial in three variables \(\lambda, x, y\), the sum over all \(G\) in that family of \(P(G,\lambda) x^{v(G)} y^{e(G)}\) where \(P(G,\lambda)\) is the chromatic polynomial of \(G\), \(v(G)\) is the number of nonrooted vertices, and \(e(G)\) the number of edges of \(G\). This concept was introduced by \textit{W. T. Tutte} for rooted planar triangulations and studied in a sequence of papers [Can. J. Math. 25, 657--671 (1973; Zbl 0268.05112), Can. J. Math. 25, 780--790 (1973; Zbl 0268.05113), Can. J. Math. 25, 929--940 (1973; Zbl 0268.05114), Can. J. Math. 26, 893--907 (1974; Zbl 0287.05103)]. In this paper, the authors study these functions for the families of rooted singular maps on the torus, projective plane, and Klein bottle, and derive some complicated expressions for the functions.