Counterexamples to the conjecture on orientations of graphs with minimum Wiener index

In Knor et al. (2016) authors conjectured that for a graph G, W-min(G) is achieved for a chi(G)-coloring-induced orientation. They also proved the conjecture holds for bipartite graphs, complete graphs, prisms and Petersen graph. However, we find that there exists a graph G such that any minimum Wiener index orientation of G is not chi(G)-coloring-induced. This means the conjecture does not hold in general. Furthermore, we explore some graphs which do not satisfy the conjecture. (C) 2017 Elsevier B.V. All rights reserved.