Sigma Partitioning: Complexity and Random Graphs

Sigma Partitioning: Complexity and Random Graphs

Blog Article

A $ extit{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 $ extit{ sigma number}$ of a graph $G$, denoted by $sigma(G)$, is the minimum number $k$ such that $ G $ has a sigma partitioning ds durga hand soap $P_1, ldots, P_k$.Also, a $ extit{ lucky labeling}$ of a graph $G$ is a function $ ell :V(G) ightarrow mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ sum_{w sim v}ell(w) eq sum_{w sim u}ell(w) $ ($ x sim y $ means that $ x $ and $y$ are adjacent).The $ extit{ lucky number}$ of $ G $, denoted by $eta(G)$, is the minimum number $k $ such that $ G $ has a lucky labeling $ ell :V(G) ightarrow mathbb{N}_k$.

It verona wig 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.