Abstract
We prove Grant's conjecture that, for all graphs G and H with H ≇ K1, the lexicographic product G[H] is semi-stable at (v, w) if H is semi-stable at w. Thus we can conclude that G[H] is semi-stable at (v, w) if and only if H is semi-stable at w.
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References
M. Behzad and G. Chartrand, "Introduction to the Theory of Graphs", Allyn and Bacon, Boston, 1971.
D.D. Grant, "Stability and operations on graphs", Lecture Notes in Mathematics No. 452, Springer-Verlag, Berlin, 1975, 116–135.
D.A. Holton, "Two applications of semi-stable graphs", Discrete Maths., 4, 1973, 151–158.
D.A. Holton and D.D. Grant, "Regular graphs and stability", J. Austral. Math. Soc., 20, Series A, 1975, 377–384.
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Holton, D.A., Stacey, K.C., McAvaney, K.L. (1977). The semi-stability of lexicographic products. In: Little, C.H.C. (eds) Combinatorial Mathematics V. Lecture Notes in Mathematics, vol 622. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069186
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DOI: https://doi.org/10.1007/BFb0069186
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