On the semi-proper orientations of graphs

Abstract

A weighted orientation of a graph G is a pair (D,w) where D is an orientation of G and w is an arc-weighting of D, that is an application A(D)→N∖{0}. The in-weight of a vertex v in a weighted orientation (D,w), denoted by S(D,w)(v), is the sum of the weights of arcs with head v in D. A semi-proper orientation is a weighted orientation such that two adjacent vertices have different in-weights. The semi-proper orientation number of a graph G, denoted by χs⃗(G), is min(D,w)∈Γmaxv∈V(G)S(D,w)(v), where Γ is the set of all semi-proper orientations of G. A semi-proper orientation (D,w) of a graph G is optimal if maxv∈V(G)S(D,w)(v)=χs⃗(G). In this work, we show that every graph G has an optimal semi-proper orientation (D,w) such that the weight of each arc is 1 or 2. We then give some bounds on the semi-proper orientation number: we show Mad(G)2≤χs⃗(G)≤Mad(G)2+χ(G)−1 and δ∗(G)+12≤χs⃗(G)≤2δ∗(G) for all graph G, where Mad(G) and δ∗(G) are the maximum average degree and the degeneracy of G, respectively. We then deduce that the maximum semi-proper orientation number of a tree is 2, of a cactus is 3, of an outerplanar graph is 4, and of a planar graph is 6. Finally, we consider the computational complexity of associated problems: we show that determining whether χs⃗(G)=χ→(G) is NP-complete for planar graphs G with χs⃗(G)=2; we also show that deciding whether χs⃗(G)≤2 is NP-complete for planar bipartite graphs G.