Note on (semi-)proper orientation of some triangulated planar graphs

A weighted orientation of a graph G is a function (D, w) with an orientation D of G and with a weight function w : E(G) -> Z(+). The in-weight w(D)(-) (v) of a vertex v in D is the value Sigma(u is an element of ND-)(v)w(uv). A weighted orientation (D, w) of G is a semi-proper orientation if w(D)(-)(v) not equal w(D)(-) (u) for all uv is an element of E(G). The semi-proper orientation number of G is defined as (chi) over right arrow (s) (G) = min((D,w)is an element of Gamma)max(v is an element of V(G)) w(D)(-)(v), where Gamma is the set of semi-proper orientations of G. When w(e) = 1 for any e is an element of E(G), this parameter is equal to the proper orientation number of G. Dehghan and Havet (2007) introduced this parameter. Inspired by Araujo et al. (2019), we want to generalize some problems in Araujo et al. (2015) about proper orientation to the semi-proper version. In this paper, we study the (semi-)proper orientation number of some triangulated planar graphs. (c) 2020 Published by Elsevier Inc.