Enumeration of graph embeddings

AbstractFor a finite connected simple graph G, let Γ be a group of graph automorphisms of G. Two 2-cell embeddings ι: G → S and j: G → S of a graph G into a closed surface S (orientable or nonorientable) are congruent with respect to Γ if there are a surface homeomorphism h:S → S and a graph automorphism γϵΓ such that hoι=joγ. In this paper, we give an algebraic characterization of congruent 2-cell embeddings, from which we enumerate the congruence classes of 2-cell embeddings of a graph G into closed surfaces with respect to a group of automorphisms of G, not just the full automorphism group. Some applications to complete graphs are also discussed. As an orientable case, the oriented congruence of a graph G into orientable surfaces with respect to the full automorphism group of G was enumerated by Mull et al. (1988)

Graph curves

AbstractWe study a family of stable curves defined combinatorially from a trivalent graph. Most of our results are related to the conjecture of Green which relates the Clifford index of a smooth curve, an important intrinsic invariant measuring the “specialness” of the geometry of the curve, to the “resolution Clifford index,” a projective invariant defined from the canonical embedding. Thus we study the canonical linear series and its powers and identify them in terms of combinatorial data on the graph; we given combinatorial criteria for the canonical series to be base point free or very ample; we prove the analogue of Noether's theorem on the projective normality of smooth canonical curves; we define a combinatorial invariant of a graph which we conjecture to be equal to the resolution Clifford index of the associated graph curve, at least for “most” graphs; and we prove our conjecture for planar graphs and for graphs of Clifford index 0. Along the way we prove a result of some independent interest on the canonical sheaves of (not necessarily arithmetically Cohen-Macaulay) face varieties. The Appendix establishes a formula connecting the combinatorics of a trivalent graph G and the minimal degree of an admissible covering of a curve of arithmetic genus 0 by the corresponding graph curve