Abstract
ABSTRACTAn -labelling of a graph G is an assignment of non-negative integers (labels) to the vertices of G such that adjacent vertices receive labels with difference at least 2, and vertices at distance 2 or 3 receive distinct labels. The span of such a labelling is the difference between the maximum and minimum labels used, and the minimum span over all -labellings of G is called the -labelling number of G, denoted by . It was shown by King, Ras and Zhou [The -labelling problem for trees, Europ. J. Combinatorics 31 (2010), pp. 1295–1306] that every tree T has , where . In this paper, we provide some sufficient conditions for . Furthermore, we completely characterize the -labelling numbers of caterpillars with . For , we prove that the -labelling numbers of caterpillars with no vertices of degree at distance 3 or 4k+2 attain the lower bound. And we show that there always exists one tree T with two vertices of degree at distance 3 or 4k+2 attaining the upper bound for any.
Published Version
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