[Paper Review] Sabidussi Versus Hedetniemi for Three Variations of the Chromatic Number

Chris Godsil, David E. Roberson+2 more|arXiv (Cornell University)|May 23, 2013

Advanced Graph Theory Research17 references4 citations

TL;DR

This paper investigates three graph parameters—vector chromatic number ($\chi_{\text{vec}}$), Lovász $\vartheta$-function of the complement ($\bar{\vartheta}$), and quantum chromatic number ($\chi_q$)—via graph homomorphisms. It proves analogs of Sabidussi’s theorem for all three and establishes Hedetniemi’s conjecture for $\bar{\vartheta}$, while proving special cases for $\chi_{\text{vec}}$ and $\chi_q$ under additional conditions, suggesting broader validity of these analogs.

ABSTRACT

We investigate vector chromatic number, Lovasz theta of the complement, and quantum chromatic number from the perspective of graph homomorphisms. We prove an analog of Sabidussi's theorem for each of these parameters, i.e. that for each of the parameters, the value on the Cartesian product of graphs is equal to the maximum of the values on the factors. We also prove an analog of Hedetniemi's conjecture for Lovasz theta of the complement, i.e. that its value on the categorical product of graphs is equal to the minimum of its values on the factors. We conjecture that the analogous results hold for vector and quantum chromatic number, and we prove that this is the case for some special classes of graphs.

Motivation & Objective

• To extend Sabidussi’s and Hedetniemi’s theorems—originally for chromatic number—to three variants: $\chi_{\text{vec}}$, $\bar{\vartheta}$, and $\chi_q$.
• To determine whether the value of each parameter on a graph product equals the maximum (Sabidussi) or minimum (Hedetniemi) of its values on the factors.
• To investigate the behavior of these parameters under Cartesian and categorical products using homomorphism-based definitions.
• To identify classes of graphs for which the quantum and vector chromatic number analogs of Hedetniemi’s conjecture hold.
• To explore the structural and spectral conditions under which $\chi_{\text{vec}}$ and $\bar{\vartheta}$ coincide, particularly in 1-homogeneous graphs.

Proposed method

• Define $\chi_{\text{vec}}$, $\bar{\vartheta}$, and $\chi_q$ via homomorphisms into specific target graphs, treating them as relaxations of chromatic number.
• Prove that $\chi_{\text{vec}}(G \mathbin{\square} H) = \max\{\chi_{\text{vec}}(G), \chi_{\text{vec}}(H)\}$ and $\bar{\vartheta}(G \mathbin{\square} H) = \max\{\bar{\vartheta}(G), \bar{\vartheta}(H)\}$, analogously to Sabidussi’s theorem.
• Establish $\bar{\vartheta}(G \times H) = \min\{\bar{\vartheta}(G), \bar{\vartheta}(H)\}$, proving the $\bar{\vartheta}$-analog of Hedetniemi’s conjecture.
• Use spectral properties of 1-homogeneous graphs (e.g., edge-transitive graphs) to derive explicit formulas for $\chi_{\text{vec}}$ and $\bar{\vartheta}$ in terms of eigenvalues.
• Introduce quantum homomorphisms and use them to prove $\chi_q(G \mathbin{\square} H) = \max\{\chi_q(G), \chi_q(H)\}$, the quantum analog of Sabidussi’s theorem.
• Prove that $\chi_q(G \times H) = \min\{\chi_q(G), \chi_q(H)\}$ when $\chi_q(G) = \bar{\vartheta}(G)$ and $\chi_q(H) = \bar{\vartheta}(H)$, using the $\bar{\vartheta}$-Hedetniemi result.

Experimental results

Research questions

• RQ1 Does the vector chromatic number satisfy a Sabidussi-type theorem on the Cartesian product of graphs?
• RQ2 Does the Lovász $\vartheta$-function of the complement satisfy a Hedetniemi-type theorem on the categorical product of graphs?
• RQ3 For which classes of graphs does the quantum chromatic number satisfy the Hedetniemi conjecture analog?
• RQ4 Can the $\chi_{\text{vec}}$-analog of Hedetniemi’s conjecture be proven in general, or only under specific spectral conditions?
• RQ5 What is the relationship between $\chi_{\text{vec}}$ and $\bar{\vartheta}$ in 1-homogeneous graphs, and does this imply equality of the parameters?

Key findings

• The paper proves that $\chi_{\text{vec}}(G \mathbin{\square} H) = \max\{\chi_{\text{vec}}(G), \chi_{\text{vec}}(H)\}$, establishing the vector chromatic number analog of Sabidussi’s theorem.
• It establishes that $\bar{\vartheta}(G \times H) = \min\{\bar{\vartheta}(G), \bar{\vartheta}(H)\}$, confirming the $\bar{\vartheta}$-analog of Hedetniemi’s conjecture.
• For 1-homogeneous graphs, $\chi_{\text{vec}}(G) = \bar{\vartheta}(G)$, and the $\chi_{\text{vec}}$-analog of Hedetniemi’s conjecture holds, as shown via eigenvalue-based formulas.
• The quantum chromatic number satisfies $\chi_q(G \mathbin{\square} H) = \max\{\chi_q(G), \chi_q(H)\}$, proving the quantum analog of Sabidussi’s theorem.
• For graphs where $\chi_q(G) = \bar{\vartheta}(G)$, the quantum analog of Hedetniemi’s conjecture holds: $\chi_q(G \times H) = \min\{\chi_q(G), \chi_q(H)\}$.
• The result is applied to the graphs $\Omega_n$, showing $\chi_q(\Omega_m \times \Omega_n) = \min\{\chi_q(\Omega_m), \chi_q(\Omega_n)\}$ for all $m,n \in \mathbb{N}$.