Circular Game Coloring of Signed Graphs

Abstract

We extend the theory of circular game chromatic numbers to signed graphs by defining the invariant $\chi_c^g(G,\sigma)$ for signed graphs $(G,\sigma)$. Our analysis establishes tight bounds dependent on the structural properties of the underlying graph $G$ and its signature $\sigma$. Building on the foundational framework of Lin and Zhu \cite{LinZhu2009}, we demonstrate that the circular game chromatic number of a balanced signed graph $(G, \sigma)$ equals that of its underlying graph $G$, i.e., $\chi_c^g(G,\sigma) = \chi_c^g(G)$. For antibalanced signed graphs, we prove that $\chi_c^g(G,\sigma)$ does not exceed the chromatic number of $G$ plus one, with tightness demonstrated for odd cycles. A dichotomy emerges for bipartite graphs: $\chi_c^g(G,\sigma)$ 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.