On the semiproper orientations of graphs
Abstract
A {\it semiproper orientation} of a given graph $G$ is a function $(D,w)$ that assigns an orientation $D(e)$ and a positive integer weight $ w(e)$ to each edge $e$ such that for every two adjacent vertices $v$ and $u$, $S_{(D,w)}(v) \neq S_{(D,w)}(u) $, where $S_{(D,w)}(v) $ is the sum of the weights of edges with head $v$ in $D$. The {\it semiproper orientation number} of a graph $G$, denoted by $\overrightarrow{\chi}_s (G)$, is $ \min_{(D,w)\in \Gamma} \max_{v\in V(G)} S_{(D,w)}(v) $, where $\Gamma$ is the set of all semiproper orientations of $G$. The {\it optimal semiproper orientation} is a semiproper orientation $(D,w)$ such that $ \max_{v\in V(G)} S_{(D,w)}(v)= \overrightarrow{\chi}_s (G) $. In this work, we show that every graph $G$ has an optimal semiproper orientation $(D,w)$ such that the weight of each edge is one or two. Next, we show that determining whether a given planar graph $G$ with $\overrightarrow{\chi}_s (G)=2 $ has an optimal semiproper orientation $(D,w)$ such that the weight of each edge is one is NPcomplete. Finally, we prove that the problem of determining the semiproper orientation number of planar bipartite graphs is NPhard.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.02867
 Bibcode:
 2019arXiv190502867D
 Keywords:

 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics