The goal of the paper is to study vertices of degree 6 of minimal and contraction critical 6-connected graph, i.e., a 6-connected graph that looses 6-connectivity both upon removal and upon contraction of any edge. It is proved that if x and z are adjacent vertices of degree 6, then x and z have at least 4 common neighbors. In addition, a detailed description of the neighborhood of the set {x, z} is given. An infinite series of examples of minimal and contraction critical 6-connected graphs for which the fraction of vertices of degree 6 equals \( \frac{11}{17} \) is constructed.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 488, 2019, pp. 143–167.
Translated by A. V. Pastor.
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Pastor, A.V. On Vertices of Degree 6 of Minimal and Contraction Critical 6-Connected Graph. J Math Sci 255, 88–102 (2021). https://doi.org/10.1007/s10958-021-05351-0
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DOI: https://doi.org/10.1007/s10958-021-05351-0