Potential continuity of colorings

Perfect $2$-colorings of the Platonic graphs

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

On the Defining Number of (2n - 2)-Vertex Colorings of Kn x Kn

Clopen Graphs, Inverse Limits, and Cochromatic Numbers

A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X2 without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inve...

How many delta-matroids are there?

J. Makowsky and B. Zilber (2004) showed that many variations of graph colorings, called CP-colorings in the sequel, give rise to graph polynomials. This is true in particular for harmonious colorings, convex colorings, mcct-colorings, and rainbow colorings, and many more. N. Linial (1986) showed that the chromatic polynomial χ(G;X) is #P-hard to evaluate for all but three values X = 0, 1, 2, wh...

Improved FPTAS for Multi-spin Systems

We design deterministic fully polynomial-time approximation scheme (FPTAS) for computing the partition function for a class of multi-spin systems, extending the known approximable regime by an exponential scale. As a consequence, we have an FPTAS for the Potts models with inverse temperature β up to a critical threshold |β| = O( 1 ∆ ) where ∆ is the maximum degree, confirming a conjecture in [1...