Circular chromatic number of Cartesian product of signed graphs
Ebode Atangana Pie Desire
TL;DR
This work studies how circular coloring behaves for Cartesian products of signed graphs, introducing two product notions: Type 1, which inherits edge signs directly, and Type 2, which multiplies edge signs by vertex signs. The authors prove an exact formula for Type 1: χ_c((G,σ)□(H,τ)) = max{χ_c(G,σ), χ_c(H,τ)}, and establish a tight upper bound for Type 2: χ_c((G,σ)□'(H,τ)) ≤ 2 max{χ_c(G), χ_c(H)}, with asymptotic tightness shown via high-girth constructions. Key techniques include switching invariance, layer decompositions, and the use of digon graphs to relate signed and unsigned coloring, along with the tight-cycle framework for χ_c. The results illuminate fundamental differences between the two product types, connect to homomorphism and flow perspectives, and open multiple directions for exact values, other graph products, and algorithmic questions in the area of signed graph colorings.
Abstract
This paper studies the circular coloring of signed graphs. A signed graph is a graph with a signature that assigns a sign to each edge, either positive or negative. This paper studies circular coloring and a circular chromatic number of Type 1 and Type Cartesian products. We shall prove the following results: The circular chromatic number of Cartesian product Type 1 $(G,σ)\Box (H,τ)$ is $χ_{c}(G \Box H,σ\Boxτ)=\max\{χ_{c}(G,σ),χ_{c}(H,τ)\}$ and the circular chromatic number of Cartesian product Type 2 $(G,σ)\Box' (H,τ)$ satisfies $χ_{c}(G \Box' H,σ\Box' τ)\leq 2\max\{χ_{c}(G),χ_{c}(H)\}$.