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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=(d1,…,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) d1+…+dn≥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 d1+…+dn≥2n. (A wreath is a connected graph that has exactly one cycle and all vertices not in the cycle monovalent.)

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

References

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