Abstract
A graphH divides a graphG, writtenH|G, ifG isH-decomposable. A graphG without isolated vertices is a greatest common divisor of two graphsG 1 andG 2 ifG is a graph of maximum size for whichG|G 1 andG|G 2, while a graphH without isolated vertices is a least common multiple ofG 1 andG 2 ifH is a graph of minimum size for whichG 1|H andG 2|H. It is shown that every two nonempty graphs have a greatest common divisor and least common multiple. It is also shown that the ratio of the product of the sizes of a greatest common divisor and least common multiple ofG 1 andG 2 to the product of their sizes can be arbitrarily large or arbitrarily small. Sizes of least common multiples of various pairsG 1,G 2 of graphs are determined, including when one ofG 1 andG 2 is a cycle of even length and the other is a star.
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G. C's research was supported in part by the Office of Naval Research, under Grant N00014-91-I-1060
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Chartrand, G., Holley, L., Kubicki, G. et al. Greatest common divisors and least common multiples of graphs. Period Math Hung 27, 95–104 (1993). https://doi.org/10.1007/BF01876635
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DOI: https://doi.org/10.1007/BF01876635