Antibandwidth of Bipartite Graphs

Let G = (V,E) be a graph with n vertices and f : V (G) → {1, 2, . . . , n} be a bijective function. We define the minimum edge difference as ab(G, f) = min{|f(u)− f(v)| : (u, v) ∈ E}. We say that f is a k-antibandwidth labeling on G if ab(G, f) ≥ k. Let ab(G) = maxf{ab(G, f)}. We therefore investigate the lower bound of ab(G). In other words, to what extent can we maximize the minimum edge difference in G among all possible labelings? We first give a lower bound bn/3∆(G)c on ab(G, f). In addition, we conjecture that for any graph G with n > 2 · ∆(G), we have ab(G) ≥ bn/∆(G)c, where ∆(G) is the maximum degree of G, and show that our conjecture holds for all connected bipartite graphs with more than 2 · ∆(G) vertices. Besides solving the problem of finding the maximized minimum-difference labeling, we further explore the relationships between the antibandwidth problems and the equitable coloring problems.