Circular Game Coloring of Signed Graphs

TL;DR

This work extends the circular game chromatic framework to signed graphs by defining χ_c^g(G,σ) and establishing tight connections with balance and antibalance through switching (Zaslavsky). Balanced signatures recover the unsigned behavior χ_c^g(G,σ)=χ_c^g(G), while antibalanced signatures introduce at most one color of slack, with odd cycles achieving tightness. For bipartite graphs, a clear dichotomy emerges: χ_c^g(G,σ)=2 when balanced and ≤3 otherwise, reflecting the influence of signs on color constraints. The paper also develops polynomial-time switching tests and canonical forms, analyzes specific classes (cycles, complete graphs, trees), and provides algorithmic and complexity results, opening several avenues for future work including planar graphs and exact computational complexity questions.

Abstract

We extend the theory of circular game chromatic numbers to signed graphs by defining the invariant $χ_c^g(G,σ)$ for signed graphs $(G,σ)$. Our analysis establishes tight bounds dependent on the structural properties of the underlying graph $G$ and its signature $σ$. Building on the foundational framework of Lin and Zhu \cite{LinZhu2009}, we demonstrate that the circular game chromatic number of a balanced signed graph $(G, σ)$ equals that of its underlying graph $G$, i.e., $χ_c^g(G,σ) = χ_c^g(G)$. For antibalanced signed graphs, we prove that $χ_c^g(G,σ)$ does not exceed the chromatic number of $G$ plus one, with tightness demonstrated for odd cycles. A dichotomy emerges for bipartite graphs: $χ_c^g(G,σ)$ equals $2$ when the graph is balanced, and otherwise remains bounded above by $3$. These results rely on switching equivalence principles (Lemma \ref{lem:Zaslavsky}) and critical properties of fundamental cycles (Lemma \ref{lem:ForcingTree}), adapting classical techniques from unsigned graph theory to the signed context. We further highlight open questions regarding computational complexity and planar graph extensions, creating new bridges between combinatorial game theory and signed graph structural analysis.

Theorems & Definitions (48)

• Proposition 1: Harary's Balance Theorem
• proof
• Lemma 2: Zaslavsky Z82b
• Lemma 3
• Lemma 4
• proof
• Proposition 5
• Proposition 6
• proof : Proof Sketch
• Remark 7