Counting Homomorphisms to $K_4$-Minor-Free Graphs, Modulo 2

Counting Homomorphisms to Trees Modulo a Prime

Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of #pHomsToH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating t...

Homomorphisms of triangle-free graphs without a K5-minor

In the course of extending Grötzsch’s theorem, we prove that every triangle-free graph without a K5-minor is 3-colorable. It has been recently proved that every triangle-free planar graph admits a homomorphism to the Clebsch graph. We also extend this result to the class of triangle-free graphs without a K5-minor. This is related to some conjectures which generalize the Four-Color Theorem. Whil...

Homomorphisms of 2-Edge-Colored Triangle-Free Planar Graphs

In this article, we introduce and study the properties of some target graphs for 2-edge-colored homomorphism. Using these properties, we obtain in particular that the 2-edge-colored chromatic number of the class of triangle-free planar graphs is at most 50. We also show that it is at least 12. C © 2016 Wiley Periodicals, Inc. J. Graph Theory 00: 1–20, 2016

Counting List Homomorphisms and Graphs with Bounded Degrees

Counting Claw-Free Cubic Graphs

Let Hn be the number of claw-free cubic graphs on 2n labeled nodes. Combinatorial reductions are used to derive a second order, linear homogeneous differential equation with polynomial coefficients whose power series solution is the exponential generating function for {Hn}. This leads to a recurrence relation for Hn which shows {Hn} to be P -recursive and which enables the sequence to be comput...