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A characterization of panconnected graphs satisfying a local ore-type condition
Department of Mathematics, University of Umeå, Umeå, Sweden / Department of Mathematical Cybernetics, Yerevan State University, Yerevan, Republic of Armenia.
Department of Mathematics, University of Umeå, Umeå, Sweden.
Department of Mathematical Cybernetics, Yerevan State University, Yerevan, Republic of Armenia.
1996 (English)In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 22, no 2, p. 95-103Article in journal (Refereed) Published
Abstract [en]

It is well known that a graph G of order p ≥ 3 is Hamilton-connected if d(u) + d(v) ≥ p + 1 for each pair of nonadjacent vertices u and v. In this paper we consider connected graphs G of order at least 3 for which d(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| + 1 for any path uwv with uv ∉ E(G), where N(x) denote the neighborhood of a vertex x. We prove that a graph G satisfying this condition has the following properties: (a) For each pair of nonadjacent vertices x, y of G and for each integer k, d(x, y) ≤ k ≤ |V(G)| − 1, there is an x − y path of length k. (b) For each edge xy of G and for each integer k (excepting maybe one k η 3,4) there is a cycle of length k containing xy.Consequently G is panconnected (and also edge pancyclic) if and only if each edge of G belongs to a triangle and a quadrangle.Our results imply some results of Williamson, Faudree, and Schelp. © 1996 John Wiley & Sons, Inc.

Place, publisher, year, edition, pages
Wiley Subscription Services, Inc., A Wiley Company , 1996. Vol. 22, no 2, p. 95-103
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Mathematics
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URN: urn:nbn:se:liu:diva-143770DOI: 10.1002/(SICI)1097-0118(199606)22:2<95::AID-JGT1>3.0.CO;2-FOAI: oai:DiVA.org:liu-143770DiVA, id: diva2:1166935
Available from: 2017-12-17 Created: 2017-12-17 Last updated: 2017-12-17Bibliographically approved

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Asratian, Armen S.

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