On Domination Exponents for Pairs of Graphs

TL;DR

This paper investigates the homomorphism density domination exponent between pairs of graphs, providing exact values for certain graph pairs and revealing the complexity of characterizing these exponents across all connected graphs.

Contribution

It introduces the homomorphism density domination exponent, analyzes its properties, and computes exact and asymptotic values for specific classes of graph pairs.

Findings

Infinitely many graph families are needed to realize the domination exponent for all connected graphs.

Exact values are obtained when $H_1$ is an even cycle and $H_2$ contains a Hamiltonian cycle.

Asymptotically sharp bounds are provided for pairs of odd cycles.

Abstract

Understanding graph density profiles is notoriously challenging. Even for pairs of graphs, complete characterizations are known only in very limited cases, such as edges versus cliques. This paper explores a relaxation of the graph density profile problem by examining the homomorphism density domination exponent $C(H_1, H_2)$. This is the smallest real number $c \geq 0$ such that $t(H_1, T) \geq t(H_2, T)^c$ for all target graphs $T$ (if such a $c$ exists) where $t(H,T)$ is the homomorphism density from $H$ to $T$. We demonstrate that infinitely many families of graphs are required to realize $C(H_1, H_2)$ for all connected graphs $H_1$, $H_2$. We derive the homomorphism density domination exponent for a variety of graph pairs, including paths and cycles. As a couple of typical examples, we obtain exact values when $H_1$ is an even cycle and $H_2$ contains a Hamiltonian cycle, and provide asymptotically sharp bounds when both $H_1$ and $H_2$ are odd cycles.