In the square of graphs, hamiltonicity and pancyclicity, Hamiltonian connectedness and panconnectedness are equivalent concepts
 Doz. Dr. H. Fleischner
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
The squareG ^{2} of a graphG has the same point set asG, and two points ofG ^{2} are adjacent inG ^{2} if and only if their distance inG is at most two. The result thatG ^{2} is Hamiltonian ifG is twoconnected, has been established early in 1971. A conjecture (ofA. Bondy) followed immediately: SupposeG ^{2} to have a Hamiltonian cycle; is it true that for anyv∈V(G), there exist cyclesC _{ j } containingv and having arbitrary lengthj, 3≤j≤V(G). The proof of this conjecture is one of the two main results of this paper. The other main result states that ifG ^{2} contains a Hamiltonian pathP(v, w) joining the pointsv andw, thenG ^{2} contains for anyj withd _{ G } ^{2} (v, w)≤j≤≤V(G)−1 a pathP _{ j } (v, w) of lengthj joiningv andw. By this, a conjecture ofF. J. Faudree andR. H. Schelp is proved and generalized for the square of graphs.
However, to prove these two results extensive preliminary work is necessary in order to make the proof of the main results transparent (Theorem 1 through 5); and Theorem 3 plays a central role for the main results. As can be seen from the statement of Theorem 3, the following known results follow in a stronger form: (a) IfG is twoconnected, thenG ^{2} is Hamiltonianconnected; (b) IfG is twoconnected, thenG ^{2} is 1Hamiltonian.
 Dirac, G. A.: Minimally twoconnected graphs. J. Reine u. Angew. Math.228, 204–216 (1967).
 Faudree, F. J., andR. H. Schelp: The square of a block is strongly path connected. J. Comb. Theory, Ser. B20, 47–61 (1976).
 Fleischner, H.: On spanning subgraphs of a connected bridgeless graph and their application to DTGraphs. J. Comb. Theory, Ser. B16, 17–28 (1974).
 Fleischner, H.: The square of every twoconnected graph is Hamiltonian. J. Comb. Theory, Ser. B16, 29–34 (1974).
 Fleischner, H., andA. M. Hobbs: A necessary condition for the square of a graph to be Hamiltonian. J. Comb. Theory, Ser. B19, 97–118 (1975).
 Fleischner, H., andA. M. Hobbs: Hamiltonian total graphs. Math. Nachr.68, 59–82 (1975).
 Fleischner, H., undH. V. Kronk: Hamiltonsche Linien im Quadrat brückenloser Graphen mit Artikulationen. Mh. Math.76, 112–117 (1972).
 Harary, F.: Graph Theory. Reading, Mass.: Addison Wesley. 1969.
 Hobbs, A. M.: The square of a block is vertex pancyclic. J. Comb. Theory, Ser. B20, 1–4 (1976).
 Title
 In the square of graphs, hamiltonicity and pancyclicity, Hamiltonian connectedness and panconnectedness are equivalent concepts
 Journal

Monatshefte für Mathematik
Volume 82, Issue 2 , pp 125149
 Cover Date
 19760601
 DOI
 10.1007/BF01305995
 Print ISSN
 00269255
 Online ISSN
 14365081
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Doz. Dr. H. Fleischner ^{(1)}
 Author Affiliations

 1. Institut für Informationsverarbeitung, Fleischmarkt 20, A1010, Wien, Austria