The 3‐connected graphs having a longest cycle containing only three contractible edges

Aldred, R. E. L.; Hemminger, Robert L.; Ota, Katsuhiro

doi:10.1002/jgt.3190170310

Cited by 11 publications

(2 citation statements)

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“…Later, Ellingham et al [11] proved that for any non‐Hamiltonian 3‐connected graph, every longest cycle contains at least six contractible edges. Aldred et al [1] characterized all 3‐connected graphs that have a longest cycle containing exactly three contractible edges. Fujita [13, 16] classified all 3‐connected graphs that have a longest cycle containing precisely four contractible edges.…”

Section: Introductionmentioning

confidence: 99%

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Contractible edges in longest cycles

Chan

Kriesell

Schmidt

2023

Journal of Graph Theory

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This paper investigates the number of contractible edges in a longest cycle C $C$ of a k $k$‐connected graph ( k ≥ 3 ) $(k\ge 3)$ that is triangle‐free or has minimum degree at least 3 2 k − 1 $\frac{3}{2}k-1$. We prove that, except for two graphs, C $C$ contains at least min { | E ( C ) | , 6 } $\min \{|E(C)|,6\}$ contractible edges. For triangle‐free 3‐connected graphs, we show that C $C$ contains at least min { | E ( C ) | , 7 } $\min \{|E(C)|,7\}$ contractible edges, and characterize all graphs having a longest cycle containing exactly six/seven contractible edges. Both results are tight. Lastly, we prove that every longest cycle C $C$ of a 3‐connected graph of girth at least 5 contains at least | E ( C ) | 12 $\frac{|E(C)|}{12}$ contractible edges.

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