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

An orientation D of G is proper if for every xy is an element of E(G), we have d(D)(-)(x)not equal 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, pbon(G)<= 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 pon(T)<= 4 extends to the tight bound pbon(T)<= 4 and for cacti G, the tight bound pon(G)<= 7 extends to the tight bound pbon(G)<= 7. We also prove that there is an infinite number of trees T for which pbon(T)<pon(T). Let (H, w) be a weighted digraph with a weight function w:A(H)-> Z(+). The in-weightwH-(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)-> Z(+), such that the in-weight of any adjacent vertices are distinct and w(D)(-)(v)<= p for every v is an element of 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 spon(G)<= pbon(G) and characterize graphs G for which spon(G)=pbon(G).