## Ali Dehghan ; Mohammad-Reza Sadeghi ; Arash Ahadi - Sigma Partitioning: Complexity and Random Graphs

dmtcs:1534 - Discrete Mathematics & Theoretical Computer Science, December 17, 2018, vol. 20 no. 2 - https://doi.org/10.23638/DMTCS-20-2-19
Sigma Partitioning: Complexity and Random GraphsArticle

Authors: Ali Dehghan ; Mohammad-Reza Sadeghi ; Arash Ahadi

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into 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 of neighbors 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 a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ and $y$ are adjacent). The $\textit{ lucky number}$ of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a lucky labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured 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-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.

Volume: vol. 20 no. 2
Section: Graph Theory
Published on: December 17, 2018
Accepted on: November 22, 2018
Submitted on: July 18, 2016
Keywords: Mathematics - Combinatorics,Computer Science - Computational Complexity

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