Proper orientation, proper biorientation and semi-proper orientation numbers of graphs

Abstract

An orientation D of G is proper if for every $xy\in E(G)$, we have $d^-_D(x)\ne d^-_D(y)$. An orientation D is a p-orientation if the maximum in-degree of a vertex in D is at most p. The minimum integer p such that G has a proper p-orientation is called the proper orientation number pon(G) of G [introduced by Ahadi and Dehghan (Inf Process Lett 113:799–803, 2013)]. We introduce a proper biorientation of G, where an edge xy of G can be replaced by either arc xy or arc yx or both arcs xy and yx. Similarly to pon(G), we can define the proper biorientation number pbon(G) of G using biorientations instead of orientations. Clearly, $\textrm{pbon}(G)\le \textrm{pon}(G)$ for every graph G. We compare pbon(G) with pon(G) for various classes of graphs. We show that for trees T,  the tight bound $\textrm{pon}(T)\le 4$ extends to the tight bound $\textrm{pbon}(T)\le 4$ and for cacti G, the tight bound $\textrm{pon}(G)\le 7$ extends to the tight bound $\textrm{pbon}(G)\le 7.$ We also prove that there is an infinite number of trees T for which $\textrm{pbon}(T)< \textrm{pon}(T).$ Let (H, w) be a weighted digraph with a weight function $w: A(H)\rightarrow {\mathbb {Z}}_+.$ The in-weight $w^-_H(v)$ of a vertex v of H is the sum of the weights of arcs towards v. A semi-proper p-orientation (D, w) of an undirected graph G is an orientation D of G together with a weight function $w: A(D)\rightarrow {\mathbb {Z}}_+$, such that the in-weight of any adjacent vertices are distinct and $w^-_D(v)\le p$ for every $v\in V(D)$. The semi-proper orientation number spon(G) of a graph G (introduced by Dehghan and Havet in 2021) is the minimum p such that G has a semi-proper p-orientation (D, w) of G. We prove that $\textrm{spon}(G)\le \textrm{pbon}(G)$ and characterize graphs G for which $\textrm{spon}(G)= \textrm{pbon}(G).$