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Back in 1922, Franklin proved that every 3-polytope $$P_5$$ P 5 with minimum degree 5 has a 5-vertex adjacent to two vertices of degree at most 6, which is tight. This result has been extended and refined in several directions. In particular, Jendrol’ and Madaras (Discuss Math Graph Theory 16:207–217, 1996) ensured a 4-path with the degree sum at most 23. A path $$v_1\ldots v_k$$ ...
It is trivial that every 3-polytope has a face of degree at most 5, called minor. Back in 1940, Lebesgue gave an approximate description of minor faces in 3-polytopes. Borodin (Diskretn Anal Issled Oper 9(3), 29–35, 2002) improved Lebesgue’s description on six parameters and suggested to find a tight description of minor faces. By now, such a tight description has been obtained only for several restricted...
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