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Some panconnected and pancyclic properties of graphs with 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 Mathematical Cybernetics, Yerevan State University, Yerevan, Republic of Armenia.
1996 (English)In: Graphs and Combinatorics, ISSN 0911-0119, E-ISSN 1435-5914, Vol. 12, no 3, p. 209-219Article in journal (Refereed) Published
Abstract [en]

Asratian and Khachatrian proved that a connected graphG of order at least 3 is hamiltonian ifd(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| for any pathuwv withuv ∉ E(G), whereN(x) is the neighborhood of a vertexx.

We prove that a graphG with this condition, which is not complete bipartite, has the following properties:

  1. a) For each pair of verticesx, y with distanced(x, y) ≥ 3 and for each integern, d(x, y) ≤ n ≤ |V(G)| − 1, there is anx − y path of lengthn. 
  2. (b)For each edgee which does not lie on a triangle and for eachn, 4 ≤ n ≤ |V(G)|, there is a cycle of lengthn containinge. 
  3. (c)Each vertex ofG lies on a cycle of every length from 4 to |V(G)|. 

This implies thatG is vertex pancyclic if and only if each vertex ofG lies on a triangle.

Place, publisher, year, edition, pages
1996. Vol. 12, no 3, p. 209-219
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-143289DOI: 10.1007/BF01858455OAI: oai:DiVA.org:liu-143289DiVA, id: diva2:1161679
Available from: 2017-11-30 Created: 2017-11-30 Last updated: 2017-11-30Bibliographically approved

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