ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 19(1) (1999) 13-29
DOI: 10.7151/dmgt.1082


Ralph Faudree

Department of Mathematical Sciences
University of Memphis, Memphis, TN 38152

András Gyárfás

Computer and Automation Institute
Hungarian Academy of Sciences
Budapest, Hungary


Let C denote the claw K1,3, N the net (a graph obtained from a K3 by attaching a disjoint edge to each vertex of the K3), W the wounded (a graph obtained from a K3 by attaching an edge to one vertex and a disjoint path P3 to a second vertex), and Zi the graph consisting of a K3 with a path of length i attached to one vertex. For k a fixed positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY-free (does not contain an induced copy of C or of Y) will be determined for Y a connected subgraph of either P6, N, W, or Z3. It should be noted that the pairs of graphs CY are precisely those forbidden pairs that imply that any 2-connected graph of order at least 10 is hamiltonian. These extremal numbers give one measure of the relative strengths of the forbidden subgraph conditions that imply a graph is hamiltonian.


[1] P. Bedrossian, Forbidden subgraph and minimum degree conditions for hamiltonicity, Ph.D Thesis, Memphis State University, 1991.
[2] J.A. Bondy and U.S.R. Murty, Graph Theory With Applications (Macmillan, London and Elsevier, New York, 1976).
[3] G. Chartrand and L. Lesniak, Graphs and Digraphs (2nd ed., Wadsworth and Brooks/Cole, Belmont, 1986).
[4] G. Dirac, Some Theorems on Abstract Graphs, Proc. London Math. Soc. 2 (1952) 69-81, doi: 10.1112/plms/s3-2.1.69.
[5] P. Erdős, R.J. Faudree, C.C. Rousseau and R.H. Schelp, On Cycle Complete Graph Ramsey Numbers, J. Graph Theory 2 (1978) 53-64, doi: 10.1002/jgt.3190020107.
[6] R.J. Faudree, Forbidden Subgraphs and Hamiltonian Properties - A Survey, Congressus Numerantium 116 (1996) 33-52.
[7] R.J. Faudree, E. Flandrin and Z. Ryjácek, Claw-free Graphs - A Survey, Discrete Math. 164 (1997) 87-147, doi: 10.1016/S0012-365X(96)00045-3.
[8] R.J. Faudree and R.J. Gould, Characterizing Forbidden Pairs for Hamiltonian Properties, Discrete Math. 173 (1977) 45-60, doi: 10.1016/S0012-365X(96)00147-1.
[9] J.K. Kim, The Ramsey number R(3,t) has order of magnitude t2/logt, Random Structures Algorithms 7 (1995) 173-207, doi: 10.1002/rsa.3240070302.
[10] O. Ore, Note on Hamiltonian Circuits, Amer. Math. Monthly 67 (1960) 55, doi: 10.2307/2308928.

Received 16 February 1998
Revised 8 December 1998