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

2020/08/30 by Carol T. Zamfirescu, Zamfirescu, Carol T.
Computer Science · Engineering · Mathematics · #05C10 #05C38 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #graph theory and CDMA systems #math.CO #msc:05C10 #msc:05C38

paper · pdf · doi:10.48550/arxiv.2009.00423

3 pages, 2 figures

arxiv created 2020/08/30 · openalex publication_date 2020/08/30 · arxiv updated 2020/09/02 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

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.

Related

• Tight gaps in the cycle spectrum of 3-connected planar graphs
• Gaps in the cycle spectrum of 3-connected cubic planar graphs
• Erdös-Gyárfás Conjecture for Cubic Planar Graphs
• Longest cycles and their chords
• Long cycles and spectral radii in planar graphs