Sigma Partitioning: Complexity and Random Graphs

Abstract

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the verticesinto sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and$v$ there is an index $i$ such that $u$ and $v$ have different numbers ofneighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by$\sigma(G)$, is the minimum number $k$ such that $G$ has a sigma partitioning$P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is afunction $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacentvertices $v$ and $u$ of $G$, \sum_{w \sim v}\ell(w)\neq \sum_{w \simu}\ell(w) ($x \sim y$ means that $x$ and $y$ are adjacent). The $\textit{lucky number}$ of $G$, denoted by $\eta(G)$, is the minimum number $k$ suchthat $G$ has a lucky labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. It wasconjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $\mathbf{NP}$-complete to decide whether $\eta(G)=2$ for a given 3-regulargraph $G$. In this work, we prove this conjecture. Among other results, we givean upper bound of five for the sigma number of a uniformly random graph