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Light Minor 5-Stars in 3-Polytopes with Minimum Degree 5

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Abstract

Attempting to solve the Four Color Problem in 1940, Henry Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P5 of 3-polytopes with minimum degree 5. This description depends on 32 main parameters. Not many precise upper bounds on these parameters have been obtained as yet, even for restricted subclasses in P5. Given a 3-polytope P, by w(P) denote the minimum of the maximum degree-sum (weight) of the neighborhoods of 5-vertices (minor 5-stars) in P. In 1996, Jendrol’ and Madaras showed that if a polytope P in P5 is allowed to have a 5-vertex adjacent to four 5-vertices (called a minor (5, 5, 5, 5, ∞)-star), then w(P) can be arbitrarily large. For each P* in P5 with neither vertices of degree 6 and 7 nor minor (5, 5, 5, 5, ∞)-star, it follows from Lebesgue’s Theorem that w(P*) ≤ 51. We prove that every such polytope P* satisfies w(P*) ≤ 42, which bound is sharp. This result is also best possible in the sense that if 6-vertices are allowed but 7-vertices forbidden, or vice versa; then the weight of all minor 5-stars in P5 under the absence of minor (5, 5, 5, 5, ∞)-stars can reach 43 or 44, respectively.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    O. V. Borodin & A. O. Ivanova

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  1. O. V. Borodin
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  2. A. O. Ivanova
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Correspondence to O. V. Borodin or A. O. Ivanova.

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The authors were funded by the Russian Science Foundation (Grant 16-11-10054).

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Borodin, O.V., Ivanova, A.O. Light Minor 5-Stars in 3-Polytopes with Minimum Degree 5. Sib Math J 60, 272–278 (2019). https://doi.org/10.1134/S0037446619020071

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  • DOI: https://doi.org/10.1134/S0037446619020071

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