Abstract.
An edge of a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. A k-connected graph with no k-contractible edge is said to be contraction critically k-connected. We prove that a contraction critically 5-connected graph on n vertices has at least n/5 vertices of degree 5. We also show that, for a graph G and an integer k greater than 4, there exists a contraction critically k-connected graph which has G as its induced subgraph.
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Ando, K., Kaneko, A. & Kawarabayashi, Ki. Vertices of Degree 5 in a Contraction Critically 5-connected Graph. Graphs and Combinatorics 21, 27–37 (2005). https://doi.org/10.1007/s00373-004-0591-y
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DOI: https://doi.org/10.1007/s00373-004-0591-y