Complexity results on locally-balanced \(2\)-partitions of graphs

A \emph{\(2\)-partition of a graph \(G\)} is a function \(f:V(G)\rightarrow \{0,1\}\). A \(2\)-partition \(f\) of a graph \(G\) is a \emph{locally-balanced with an open neighborhood} if for every \(v\in V(G)\), $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1.$$ A \(2\)-partition \(f^{\prime}\) of a graph \(G\) is a \emph{locally-balanced with a closed neighborhood} if for every \(v\in V(G)\), $$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=0\}\vert - \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=1\}\vert \right\vert\leq 1.$$ In this paper we prove that the problem of the existence of locally-balanced \(2\)-partition with an open (closed) neighborhood is \(NP\)-complete for some restricted classes of graphs. In particular, we show that the problem of deciding if a given graph has a locally-balanced \(2\)-partition with an open neighborhood is \(NP\)-complete for biregular bipartite graphs and even bipartite graphs with maximum degree \(4\), and the problem of deciding if a given graph has a locally-balanced \(2\)-partition with a closed neighborhood is \(NP\)-complete even for subcubic bipartite graphs and odd graphs with maximum degree \(3\). Last results prove a conjecture of Balikyan and Kamalian.