×

Hamiltonian \(N_{2}\)-locally connected claw-free graphs. (English) Zbl 1060.05061

Summary: A graph \(G\) is \(N_2\)-locally connected if for every vertex \(\upsilon\) in \(G\), the edges not incident with \(\upsilon\) but having at least one end adjacent to \(\upsilon\) in \(G\) induce a connected graph. In 1990, Z. Ryjáček [J. Graph Theory 14, 321–331 (1990; Zbl 0718.05036)] conjectured that every 3-connected \(N_2\)-locally connected claw-free graph is Hamiltonian. This conjecture is proved in this note.

MSC:

05C40 Connectivity
05C45 Eulerian and Hamiltonian graphs

Citations:

Zbl 0718.05036
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] and Graph theory with applications, Macmillan, London and Elsevier, New York, 1976. · Zbl 1226.05083 · doi:10.1007/978-1-349-03521-2
[2] Harary, Canad Math Bull 8 pp 701– (1965) · Zbl 0136.44704 · doi:10.4153/CMB-1965-051-3
[3] Lai, Discrete Math 94 pp 11– (1991)
[4] Li, Ars Combin 50 pp 279– (1998)
[5] Li, Ars Combinatoria 62 pp 281– (2002)
[6] Oberly, J Graph Theory 3 pp 351– (1979)
[7] Paulraja, Ars Combin 24 pp 57– (1987)
[8] Ryjá?ek, J Graph Theory 14 pp 321– (1990)
[9] Ryjá?ek, J Combin Theory Ser B 70 pp 217– (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.