Chords of longest circuits in locally planar graphs

Kawarabayashi, Ken-ichi; Niu, Jianbing; Zhang, Cun-Quan

doi:10.1016/j.ejc.2005.07.017

Cited by 9 publications

(5 citation statements)

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“…Here a longest cycle is a cycle of maximum length and a chord of a cycle C is an edge between two vertices of C that are nonadjacent on C. Thomassen [10] proved that his conjecture is true for cubic graphs. The conjecture has also been verified for graphs embeddable in several surfaces [5][6][7]. In particular, Zhang [11] proved that the conjecture is true for planar graphs with minimum degree at least four.…”

Section: Introductionmentioning

confidence: 83%

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

Broersma

Kang

2013

Graphs and Combinatorics

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We verify two special cases of Thomassen's conjecture of 1976 stating that every longest cycle in a 3-connected graph contains a chord. We prove that Thomassen's conjecture is true for two classes of 3-connected graphs that have a bounded number of removable edges on or off a longest cycle. Here an edge e of a 3-connected graph G is said to be removable if G − e is still 3-connected or a subdivision of a 3-connected (multi)graph. We give examples to show that these classes are not covered by previous results. Keywords 3-Connected graph • Removable edge • Chord Mathematics subject classifications (2000) 05C40 • 05C38 • 05C75

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Section: Introductionmentioning

confidence: 83%

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

Broersma

Kang

2013

Graphs and Combinatorics

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“…That conjecture has been proved for planar graphs with minimum degree at least four [9], cubic graphs [8] and graphs embeddable in several surfaces [4,5,6]. In this paper, we prove it for planar graphs in general.…”

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confidence: 79%

Every longest circuit of a 3‐connected, K_3,3‐minor free graph has a chord

Birmelé

2008

Journal of Graph Theory

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“…The first result regarding planar graphs was due to Zhang [13], who showed the conjecture holds for cubic planar graphs or planar graphs with minimum degree at least four. Subsequently, Kawarabayashi et al [5] verified the conjecture for locally 4-connected planar graphs, and Birmelé [2] verified the result for every 3-connected graph with no K 3,3 -minor. Wu et al [12] verified the result for certain classes of graphs that have a bounded number of removable edges.…”

Section: Introductionmentioning

confidence: 91%

A Cycle of Maximum Order in a Graph of High Minimum Degree has a Chord

Harvey

2017

Electron. J. Combin.

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A well-known conjecture of Thomassen states that every cycle of maximum order in a $3$-connected graph contains a chord. While many partial results towards this conjecture have been obtained, the conjecture itself remains unsolved. In this paper, we prove a stronger result without a connectivity assumption for graphs of high minimum degree, which shows Thomassen's conjecture holds in that case. This result is within a constant factor of best possible. In the process of proving this, we prove a more general result showing that large minimum degree forces a large difference between the order of the largest cycle and the order of the largest chordless cycle.

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Chords of longest circuits in locally planar graphs

Cited by 9 publications

References 12 publications

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

Every longest circuit of a 3‐connected, K_3,3‐minor free graph has a chord

A Cycle of Maximum Order in a Graph of High Minimum Degree has a Chord

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Chords of longest circuits in locally planar graphs

Cited by 9 publications

References 12 publications

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

Every longest circuit of a 3‐connected, K3,3‐minor free graph has a chord

A Cycle of Maximum Order in a Graph of High Minimum Degree has a Chord

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Every longest circuit of a 3‐connected, K_3,3‐minor free graph has a chord