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Tight super-edge-graceful labelings of trees and their applications. (English) Zbl 1346.05250
Summary: The concept of graceful labeling of graphs has been extensively studied. J. Mitchem and A. Simoson [Ars Comb. 37, 97–111 (1994; Zbl 0805.05074)] introduced a stronger concept called super-edge-graceful labeling for some classes of graphs. Among many other interesting pioneering results, J. Mitchem and A. Simoson [loc. cit.] provided a simple but powerful recursive way of constructing super-edge-graceful trees of odd order. In this note, we present a stronger concept of “tight” super-edge-graceful labeling. Such a super-edge graceful labeling has an additional constraint on the edge and vertices with the largest and smallest labels. This concept enables us to recursively construct tight super-edge-graceful trees of any order. As applications, we provide insights on the characterization of super-edge-graceful trees of diameter 4, a question posed by P. T. Chung et al. [Congr. Numerantium 181, 5–17 (2006; Zbl 1119.05095)]. We also observe infinite families of super-edge-graceful trees that can be generated from tight labelings. Given the direct applications of “tight” super-edge-graceful labeling to the study of super-edge-graceful labelings, we note that it is worthwhile to further examine recursively generated tight super-edge-graceful trees.
MSC:
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C05 Trees
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