Counterexamples to a conjecture of Merker on 3-connected cubic planar graphs with a large cycle spectrum gap

Carol T Zamfirescu

doi:10.1016/j.disc.2022.112824

Counterexamples to a conjecture of Merker on 3-connected cubic planar graphs with a large cycle spectrum gap

Carol T Zamfirescu

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Abstract

Merker conjectured that if k≥2 is an integer and G a 3-connected cubic planar graph of circumference at least k, then the set of cycle lengths of G must contain at least one element of the interval [k,2k+2]. We here prove that for every even integer k≥6 there is an infinite family of counterexamples.

Original language	English
Article number	112824
Number of pages	2
Journal	Discrete Mathematics
Volume	345
Issue number	6
DOIs	https://doi.org/10.1016/j.disc.2022.112824
Publication status	Published - Jun 2022
Externally published	Yes

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© 2022 Elsevier B.V.

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10.1016/j.disc.2022.112824

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title = "Counterexamples to a conjecture of Merker on 3-connected cubic planar graphs with a large cycle spectrum gap",

abstract = "Merker conjectured that if k≥2 is an integer and G a 3-connected cubic planar graph of circumference at least k, then the set of cycle lengths of G must contain at least one element of the interval [k,2k+2]. We here prove that for every even integer k≥6 there is an infinite family of counterexamples.",

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AB - Merker conjectured that if k≥2 is an integer and G a 3-connected cubic planar graph of circumference at least k, then the set of cycle lengths of G must contain at least one element of the interval [k,2k+2]. We here prove that for every even integer k≥6 there is an infinite family of counterexamples.

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DO - 10.1016/j.disc.2022.112824

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JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 6

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Counterexamples to a conjecture of Merker on 3-connected cubic planar graphs with a large cycle spectrum gap

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