Fractional balanced chromatic number of signed subcubic graphs
Xiaolan Hu, Luis Kuffner, Jiaao Li, Reza Naserasr, Lujia Wang, Zhouningxin Wang, Xiaowei Yu
TL;DR
This work investigates fractional balanced colorings in signed graphs, introducing the fractional balanced chromatic number $\chi_{\text{fb}}$ via $(p,q)$-colorings where each color class is balanced. The authors prove a tight bound for signed subcubic graphs: every such graph not switching equivalent to $(K_4,-)$ admits a $(5,3)$-coloring, yielding $\chi_{\text{fb}} \le \frac{5}{3}$; the exceptional $(K_4,-)$ attains this bound only in the limit with colorings of the form $(2p,p)$, and the extremal graph $\widehat{K}_4^{\bullet}$ achieves $\chi_{\text{fb}}=\frac{5}{3}$. A key part of the proof is a robust extension framework for $(5,\phi)$-colorings and a minimal-counterexample analysis that rules out multiple local configurations, enabling a complete coloring of all admissible graphs. The results connect to signed-graph Hadwiger-type questions and establish a tight, structure-driven bound, with open questions about improving the bound under forbidden subgraphs and identifying further critical families.
Abstract
A signed graph is a pair $(G,σ)$, where $G$ is a graph and $σ: E(G)\rightarrow \{-, +\}$, called signature, is an assignment of signs to the edges. Given a signed graph $(G,σ)$ with no negative loops, a balanced $(p,q)$-coloring of $(G,σ)$ is an assignment $f$ of $q$ colors to each vertex from a pool of $p$ colors such that each color class induces a balanced subgraph, i.e., no negative cycles. Let $(K_4,-)$ be the signed graph on $K_4$ with all edges being negative. In this work, we show that every signed (simple) subcubic graph admits a balanced $(5,3)$-coloring except for $(K_4,-)$ and signed graphs switching equivalent to it. For this particular signed graph the best balanced colorings are $(2p,p)$-colorings.