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Sequences Realizable by Graphs with Hamiltonian Squares

Published online by Cambridge University Press:  20 November 2018

V. Chungphaisan
V. Chungphaisan*
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada
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Abstract

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Let d=(d 1,…,dn ) be a sequence of positive integers. In this note we show that d is realizable by a graph whose square is hamiltonian if and only if (i) d is realizable by some graph, (ii) n≥3, and (iii) d 1+…+d n≥2(n-1). In fact, we prove that if d is realizable by a connected graph, then d is realizable by a graph with a spanning caterpillar. From this it follows that if d is realizable by a connected graph, it is realizable by a graph whose square is pancyclic. We also prove that d is realizable by a graph with a spanning wreath if and only if d is realizable by some graph and d 1+…+dn ≥2n. (A wreath is a connected graph that has exactly one cycle and all vertices not in the cycle monovalent.)

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 5 , February 1975 , pp. 629 - 631
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Bondy, J. A., Pancyclic graphs, Proc. Second Louisiana Conference on Combinatorics, Graph Theory and Computing, 167–172, Louisiana State University Baton Rouge, La., 1971.Google Scholar
2. Edmonds, J., Existence of k-edge connected ordinary graphs with prescribed degrees, J. Res. Nat. Bur. Standards Sect. B 68B (1964), 73–74.Google Scholar
3. Harary, F., Graph Theory. (Addison-Wesley, Reading, Mass. 1969).Google Scholar