Fractional coloring of product signed graphs

TL;DR

This work analyzes the fractional chromatic number of direct products of signed graphs, focusing on when one factor is a signed circulant graph $G(n,S,T)$. It introduces fractional persistence and restricted fractional cliques as key tools, proving that for any signed graph $(G,\sigma)$ and any signed circulant $(H,\tau)=G(n,S,T)$, the equality $\chi_f((G,\sigma)\times(H,\tau))=\min\{\chi_f(G,\sigma),\chi_f(H,\tau)\}$ holds. The authors develop a framework using fractional cliques and a multiplicative weighting $\rho$ to bound the product and show that signed circulant graphs are highly fractional persistent, enabling the main result. They further connect these findings to Hedetniemi's conjecture and discuss limitations, especially the restriction to direct products and the computational complexity of determining $\chi_f$. The study advances understanding of fractional coloring in structured signed graphs and points to broader applications and future research directions in graph products and invariant theory.

Abstract

This study examines the fractional chromatic number associated with the direct product of signed graphs. It shows that if $(H,τ)$ is a signed circulant graph $G(n,S,T)$, then for any signed graph $(G,σ)$, the fractional chromatic number of their direct product is the lower number between the fractional chromatic number of $(G,σ)$ and $(H,τ)$.

Theorems & Definitions (28)

• Theorem 2
• Lemma 3
• proof
• Definition 4
• Definition 5
• Lemma 6
• proof
• Lemma 8
• proof
• Corollary 9