Long paths need not minimize $H$-colorings among trees

TL;DR

The paper investigates whether the $n$-vertex path minimizes the number of $H$-colorings among all $n$-vertex trees, a question known to fail in general due to Leontovich-type counterexamples. It develops an automorphic-similarity framework and introduces path-like families to express $\hom(G_n,H)$ as a linear combination of eigenvalues of the quotient matrix $M(H)$, enabling precise asymptotic comparisons. The authors explicitly exhibit $H=T(18,3,32)$ that is Leontovich at $n=7$ and extend the phenomenon to infinitely many $n$, including odd and even cases, via $H=T(7,1,9)$ and $\widehat{H}=\widehat{T}(400,3,800)$, respectively. The results demonstrate that, for large $n$, there exist trees $E_n$ with strictly fewer $H$-colorings than the path $P_n$, and they raise several open problems about minimality, diameter constraints, and the smallest strongly Leontovich graphs, with a companion paper providing broader criteria.

Abstract

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. By appropriate choice of $H$, these colorings can express, for instance, the independent sets or proper vertex colorings of $G$. Sidorenko proved that for any $H$, the $n$-vertex star admits at least as many $H$-colorings as any other $n$-vertex tree, but the minimization question remains open in general. For many graphs $H$, path graphs are among the trees with the fewest $H$-colorings, but work of Leontovich and subsequently Csikvári and Lin shows that there is a graph $E_7$ on seven vertices and a target graph $H$ for which there are strictly fewer $H$-colorings of $E_7$ than of the path on seven vertices. We introduce a new strategy for enumerating homomorphisms from path-like trees to highly symmetric target graphs that allows us to make the previous observations completely explicit and extend them to infinitely many $n$ beyond $n=7$. In particular, we exhibit a target graph $H$ with the property that for each sufficiently large $n$, there is a tree $E_n$ on $n$ vertices that admits strictly fewer $H$-colorings than the path on $n$ vertices.

Figures (3)

• Figure 1: A visual representation of the graph $T(x,y,z)$.
• Figure 2: $P_7$ and $E_7$.
• Figure 3: The graphs $P_{n+4}$ (later referred to as $G_n$) and $E_{n+4}$ (later referred to as $G'_n$).

Theorems & Definitions (16)

• Theorem 1.1: Sidorenko
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof