Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hoffman-London graphs: When paths minimize $H$-colorings among trees

David Galvin, Phillip Marmorino, Emily McMillon, JD Nir, Amanda Redlich

TL;DR

The paper develops a unifying framework to identify Hoffman-London graphs—target graphs $H$ for which, among all trees on $n$ vertices, the path minimizes the number of $H$-colorings $ hom(T,H) $. Central to the approach is the automorphic similarity matrix $M(H)$ and the increasing columns property, which, via a KC-move based argument and a recursive tree-walk analysis, ensures $P_n$ minimizes $ hom(T_n,H) $ when the criterion holds. The authors apply the framework to a wide range of target graphs, proving Hoffman-London (and often strongly Hoffman-London) behavior for loop threshold graphs, paths and looped paths, blow-ups of fully looped stars (Widom-Rowlinson related models), and others, and they fully classify all $H$ with at most three vertices. They also connect these combinatorial results to statistical physics via partition functions, showing how blow-ups provide sharp bounds for hard-core, Widom-Rowlinson, and capacity models on trees. The work extends and strengthens existing results, provides new families of Hoffman-London graphs, and raises several open problems on odd-path minimizers and broader graph classes.

Abstract

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. The number of $H$-colorings of $G$, $\hom(G,H)$, has been studied for many classes of $G$ and $H$. In particular, extremal questions of maximizing and minimizing $\hom(G,H)$ have been considered when $H$ is a clique or $G$ is a tree. In this paper, we develop a new technique using automorphisms of $H$ to show that $\hom(T,H)$ is minimized by paths as $T$ varies over trees on a fixed number of vertices. We introduce the term Hoffman-London to refer to graphs that are minimal in this sense. In particular, we define an automorphic similarity matrix which is used to compute $\hom(T,H)$ and give matrix conditions under which $H$ is Hoffman-London. We then apply this technique to identify several families of graphs that are Hoffman-London, including loop threshold graphs and some with applications in statistical physics (e.g. the Widom-Rowlinson model). By combining our approach with a few other observations, we fully characterize the minimizing trees for all graphs $H$ on three or fewer vertices.

Hoffman-London graphs: When paths minimize $H$-colorings among trees

TL;DR

The paper develops a unifying framework to identify Hoffman-London graphs—target graphs for which, among all trees on vertices, the path minimizes the number of -colorings . Central to the approach is the automorphic similarity matrix and the increasing columns property, which, via a KC-move based argument and a recursive tree-walk analysis, ensures minimizes when the criterion holds. The authors apply the framework to a wide range of target graphs, proving Hoffman-London (and often strongly Hoffman-London) behavior for loop threshold graphs, paths and looped paths, blow-ups of fully looped stars (Widom-Rowlinson related models), and others, and they fully classify all with at most three vertices. They also connect these combinatorial results to statistical physics via partition functions, showing how blow-ups provide sharp bounds for hard-core, Widom-Rowlinson, and capacity models on trees. The work extends and strengthens existing results, provides new families of Hoffman-London graphs, and raises several open problems on odd-path minimizers and broader graph classes.

Abstract

Given a graph and a target graph , an -coloring of is an adjacency-preserving vertex map from to . The number of -colorings of , , has been studied for many classes of and . In particular, extremal questions of maximizing and minimizing have been considered when is a clique or is a tree. In this paper, we develop a new technique using automorphisms of to show that is minimized by paths as varies over trees on a fixed number of vertices. We introduce the term Hoffman-London to refer to graphs that are minimal in this sense. In particular, we define an automorphic similarity matrix which is used to compute and give matrix conditions under which is Hoffman-London. We then apply this technique to identify several families of graphs that are Hoffman-London, including loop threshold graphs and some with applications in statistical physics (e.g. the Widom-Rowlinson model). By combining our approach with a few other observations, we fully characterize the minimizing trees for all graphs on three or fewer vertices.
Paper Structure (12 sections, 24 theorems, 50 equations, 10 figures, 1 table)

This paper contains 12 sections, 24 theorems, 50 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary observations
  3. Proofs of Theorem \ref{['thm:ordering_framework']} and Corollary \ref{['cor:uniquenessalt']}
  4. Applications of Theorem \ref{['thm:ordering_framework']}
  5. Two previous results on Hoffman-London graphs
  6. Target graphs with two automorphic similarity classes
  7. Paths and looped paths as target graphs
  8. Loop threshold graphs
  9. Adding looped dominating vertices to a target graph
  10. Connection to statistical physics
  11. Classification of target graphs with at most three vertices
  12. Open problems

Key Result

Theorem 1.1

Fix $H$ and $n \ge 1$. For any $T_n \in {\mathcal{T}}_n$,

Figures (10)

  • Figure 1: The tensor product $K_2 \times K_2$.
  • Figure 2: Deconstructing $T$ into $T'$ and $T"$.
  • Figure 3: A demonstration of a KC move.
  • Figure 4: The Folkman graph.
  • Figure 5: A visualization of the family of graphs $H(a,b,\ell)$.
  • ...and 5 more figures

Theorems & Definitions (51)

  • Theorem 1.1: Sidorenko Sidorenko1994
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.5
  • Theorem 2.6
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 41 more