Abstract
An L(2, 1)-labelling of a finite graph \(\varGamma \) is a function that assigns integer values to the vertices \(V(\varGamma )\) of \(\varGamma \) (colouring of \(V(\varGamma )\) by \({\mathbb {Z}}\)) so that the absolute difference of two such values is at least 2 for adjacent vertices and is at least 1 for vertices, which are precisely distance 2 apart. The lambda number \(\lambda (\varGamma )\) of \(\varGamma \) measures the least number of integers needed for such a labelling (colouring). A power graph \(\varGamma _G\) of a finite group G is a graph with vertex set as the elements of G and two vertices are joined by an edge if and only if one of them is a positive integer power of the other. It is known that \(\lambda (\varGamma _G) \ge |G|\) for any finite group. In this paper, we show that if G is a finite group of a prime power order, then \(\lambda (\varGamma _G) = |G|\) if and only if G is neither cyclic nor a generalized quaternion 2-group. This settles a partial classification of finite groups achieving the lower bound of lambda number.
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The authors would like to thank the anonymous referee for the improvement of the exposition of the article.
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Sarkar, S., Mishra, M. The lambda number of the power graph of a finite p-group. J Algebr Comb 57, 101–110 (2023). https://doi.org/10.1007/s10801-022-01158-7
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DOI: https://doi.org/10.1007/s10801-022-01158-7