A graphG withn vertices has propertyp(r, s) ifG contains a path of lengthr and if every such path is contained in a circuit of lengths. G. A. Dirac and C. Thomassen [Math. Ann.203 (1973), 65–75] determined graphs with propertyp(r,r+1). We determine the least number of edges in a graphG in order to insure thatG has propertyp(r,s), we determine the least number of edges possible in a connected graph with propertyp(r,s) forr=1 and alls, forr=k ands=k+2 whenk=2, 3, 4, and we give bounds in other cases. Some resulting extremal graphs are determined. We also consider a generalization of propertyp(2,s) in which it is required that each pair of edges is contained in a circuit of lengths. Some cases of this last property have been treated previously by U. S. R. Murty [inProof Techniques in Graph Theory, ed. F. Harary, Academic Press, New York, 1969, pp. 111–118].
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Reid, K.B., Thomassen, C. Edge sets contained in circuits. Israel J. Math. 24, 305–319 (1976). https://doi.org/10.1007/BF02834761