Fractional balanced chromatic number of signed subcubic graphs

Abstract

A signed graph is a pair $(G,\sigma )$, where $G$ is a graph and $\sigma :E(G)\to \{-,+\}$, called signature, is an assignment of signs to the edges. Given a signed graph $(G,\sigma )$ with no negative loops, a balanced $(p,q)$-coloring of $(G,\sigma )$ 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.