The chromatic polynomial of a graph G, denoted P (G, m), is equal to the number of proper m-
colorings of G. The list color function of graph G, denoted P ℓ (G, m), is a list analogue of the …

This survey concerns regular graphs that are extremal with respect to the number of
independent sets and, more generally, graph homomorphisms. More precisely, in the family …

By a theorem of Johansson, every triangle-free graph G of maximum degree Δ has
chromatic number at most (C+ o (1)) Δ/log⁡ Δ for some universal constant C> 0. Using the …

We analyse uniformly random proper k-colourings of sparse graphs with maximum degree Δ
in the regime Δ< k ln k. This regime corresponds to the lower side of the shattering threshold …

We prove tight upper and lower bounds on the internal energy per particle (expected
number of monochromatic edges per vertex) in the anti‐ferromagnetic Potts model on cubic …

In this paper we study the following problem. Let $ A $ be a fixed graph, and let $\hom (G, A)
$ denote the number of homomorphisms from a graph $ G $ to $ A $. Furthermore, let $ v (G) …

Abstract The Widom–Rowlinson graph, H WR, is the fully looped path on three vertices. Let
hom (G, H WR) be the number of graph homomorphisms from G to H WR or, equivalently …

Graph theory first arose in 1736 when Euler developed the basic concepts solving the
Bridges of Konigsberg problem. Many modern areas of graph theory are unified in the study …

The goal of this book is to give an account into the study of counting various objects in
sparse graphs. The main objects whose study will run through the book are matchings …

The chromatic polynomial of a graph $ G $, denoted $ P (G, m) $, is equal to the number of
proper $ m $-colorings of $ G $. The list color function of graph $ G $, denoted $ P_ {\ell}(G …