On the in–out–proper orientations of graphs

Abstract

An orientation of a graph G is in–out–proper if any two adjacent vertices have different in–out-degrees, where the in–out-degree of each vertex is equal to the in-degree minus the out-degree of that vertex. The in–out–proper orientation number of a graph G, denoted by χ↔(G), is minD∈Γmaxv∈V(G)|dD±(v)|, where Γ is the set of in–out–proper orientations of G and dD±(v) is the in–out-degree of the vertex v in the orientation D. Borowiecki et al. proved that the in–out–proper orientation number is well-defined for any graph G (Borowiecki et al., 2012). So we have χ↔(G)≤Δ(G), where Δ(G) is the maximum degree of vertices in G. We conjecture that there exists a constant number c such that for every planar graph G, we have χ↔(G)≤c. Toward this speculation, we show that for every tree T we have χ↔(T)≤3 and this bound is sharp. Next, we study the in–out–proper orientation number of subcubic graphs. By using the properties of totally unimodular matrices we show that there is a polynomial time algorithm to determine whether χ↔(G)≤2, for a given graph G with maximum degree three. On the other hand, we show that it is NP-complete to decide whether χ↔(G)≤1 for a given bipartite graph G with maximum degree three. Finally, we study the in–out–proper orientation number of regular graphs.