Bounded degree and planar spectra

The finite spectrum of a first-order sentence is the set of positive integers that are the sizes of its models. The class of finite spectra is known to be the same as the complexity class NE. We consider the spectra obtained by limiting models to be either planar (in the graph-theoretic sense) or by bounding the degree of elements. We show that the class of such spectra is still surprisingly rich by establishing that significant fragments of NE are included among them. At the same time, we establish non-trivial upper bounds showing that not all sets in NE are obtained as planar or bounded-degree spectra

Seurat games on Stockmeyer graphs

We define a family of vertex colouring games played over a pair of graphs or digraphs (G, H) by players ∀ and ∃. These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number n such that ∀ always has a winning strategy in the game with n colours whenever G 6∼= H. This is related to the reconstruction conjecture for graphs and the degree-associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with n = 3 for graphs, and the same is true for the degree-associated reconstruction conjecture and our conjecture for digraphs. We show (for any k < ω) that the 2-colour game can distinguish certain non-isomorphic pairs of graphs that cannot be distinguished by the k-dimensional Weisfeiler-Leman algorithm. We also show that the 2-colour game can distinguish the non-isomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture

How I got to like graph polynomials

For Boris Zilber on his 75th birthday. I trace the roots of my collaboration with Boris Zilber, which combines categoricity theory, finite model theory, algorithmics, and combinatorics.Comment: 11 page