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All Tight Descriptions of 4-Paths in 3-Polytopes with Minimum Degree 5

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Abstract

Back in 1922, Franklin proved that every 3-polytope \(P_5\) 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 \(v_1\ldots v_k\) is a \((d_1,\ldots d_k)\)-path if \(d(v_i)\le d_i\) whenever \(1\le i\le k\). Recently, Borodin and Ivanova proved that every \(P_5\) has a (6, 5, 6, 6)-path or (5, 5, 5, 7)-path, and Ivanova proved the existence of a (5, 6, 6, 6)-path or (5, 5, 5, 7)-path, where both descriptions are tight. The purpose of our paper is to give all tight descriptions of 4-paths in \(P_5\)s.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    Ts. Ch.-D. Batueva, O. V. Borodin & A. O. Ivanova

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  1. Ts. Ch.-D. Batueva
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  2. O. V. Borodin
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  3. A. O. Ivanova
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Correspondence to A. O. Ivanova.

Additional information

This research was funded by the Russian Science Foundation (Grant 16-11-10054).

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Batueva, T.CD., Borodin, O.V. & Ivanova, A.O. All Tight Descriptions of 4-Paths in 3-Polytopes with Minimum Degree 5. Graphs and Combinatorics 33, 53–62 (2017). https://doi.org/10.1007/s00373-016-1755-2

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