Sigma Partitioning: Complexity and Random Graphs - Dehghan, Ali; Sadeghi, Mohammad-Reza; Ahadi, Arash (Discrete mathematics and theoretical computer science, 2018)
A $sigma partitioning$ of a graph $G$ is a partition of the vertices into sets $P_1, …, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $ sigma number$ of a graph $G$, denoted by $σ(G)$, is the minimum number $k$ such that $ G $ has a sigma partitioning $P_1, …, P_k$. Also, a $ lucky labeling$ of a graph $G$ is a function $ \ell :V(G) → \mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)≠ \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ and $y$ are adjacent). The $ lucky number$ of $ G $, denoted by $η(G)$, is the minimum number $k $ such that $ G $ has a lucky labeling $ \ell :V(G) → \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $ NP $-complete to decide whether $ η(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph.