A study on cyclic bandwidth sum

Suppose G is a graph of p vertices. A proper labeling f of G is a one-to-one mapping f:V(G)→{1,2,…,p}. The cyclic bandwidth sum of G with respect to f is defined by CBS _f (G)=∑ _uv∈E(G)|f(v)−f(u)| _p , where |x| _p =min {|x|,p−|x|}. The cyclic bandwidth sum of G is defined by CBS(G)=min {CBS _f (G): f is a proper labeling of G}. The bandwidth sum of G with respect to f is defined by BS _f (G)=∑ _uv∈E(G)|f(v)−f(u)|. The bandwidth sum of G is defined by BS(G)=min {BS _f (G): f is a proper labeling of G}. In this paper, we give a necessary and sufficient condition for BS(G)=CBS(G), and use this to show that BS(T)=CBS(T) when T is a tree. We also find cyclic bandwidth sums of complete bipartite graphs. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. Supported in part by the National Science Council under grants NSC91-2115-M-156-001.