Abstract
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.
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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.
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Chen, YD., Yan, JH. A study on cyclic bandwidth sum. J Comb Optim 14, 295–308 (2007). https://doi.org/10.1007/s10878-007-9051-y
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DOI: https://doi.org/10.1007/s10878-007-9051-y