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1-Tough cocomparability graphs are hamiltonian

✍ Scribed by Jitender S. Deogun; Dieter Kratsch; George Steiner

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 413 KB
Volume: 170
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(95)00359-5

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We prove that every 18-tough chordal graph has a Hamiltonian cycle.

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We prove the result stated in the title. Furthermore, it is proved that for any > 0, there is a 1-tough chordal planar graph G such that the length of a longest cycle of G is less than |V (G )|.

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By a theorem of the toughness t(G) of a non-hamiitonian maximal planar graph G is less than or equal to 2. Improving a result of , it is shown that the shortness exponent of the class of maximal planar graphs with toughness greater than or equal to ~ is less than 1.

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Bauer, Morgana, Schmeichel and Veldman have conjectured that the circumference c(G) of any 1-tough graph G of order n >t 3 with minimum degree 6 >/n/3 is at least min{n,(3n+ 1)/4+6/2} ~>(lln+ 3)/12. They proved that under these conditions, c(G)>~min{n,n/2+6}>15n/6. Then Bauer, Schmeichei and Veldman

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A graph is k-triangular if each edge is in at least k triangles. Triangular is a synonym for l-triangular. It is shown that the line graph of a triangular graph of order at least 4 is panconnected if and only if it is 3-connected. Furthermore, the line graph of a k-triangular graph is k-harniltonian