PDF
Discussiones Mathematicae Graph Theory 32(1) (2012)
153-160
DOI: https://doi.org/10.7151/dmgt.1593
On a Generalization of the Friendship Theorem
Mohammad Hailat
Department of Mathematical Sciences |
Abstract
The Friendship Theorem states that if any two people, of a group of at least three people, have exactly one friend in common, then there is always a person who is everybody's friend. In this paper, we generalize the Friendship Theorem to the case that in a group of at least three people, if every two friends have one or two common friends and every pair of strangers have exactly one friend then there exist one person who is friend to everybody in the group. In particular, we show that the graph corresponding to this problem is of type G = K1 ∨(sK2+ tK3), where s and t are non-negative integers and Km is the complete graph on m vertices.
Keywords: (λ,μ)-graph, Friendship Theorem
2010 Mathematics Subject Classification: 05C75.
References
[1] | J. Bondy, Kotzig's Conjecture on generalized friendship graphs --- a survey, Annals of Discrete Mathematics 27 (1985) 351--366. |
[2] | P. Erdös, A. Rènyi and V. Sós, On a problem of graph theory, Studia Sci. Math 1 (1966) 215--235. |
[3] | R. Gera and J. Shen, Extensions of strongly regular graphs, Electronic J. Combin. 15 (2008) # N3 1--5. |
[4] | J. Hammersley, The friendship theorem and the love problem, in: Surveys in Combinatorics, London Math. Soc., Lecture Notes 82 (Cambridge University Press, Cambridge, 1989) 127--140. |
[5] | N. Limaye, D. Sarvate, P. Stanika and P. Young, Regular and strongly regular planar graphs, J. Combin. Math. Combin. Compt 54 (2005) 111--127. |
[6] | J. Longyear and T. Parsons, The friendship theorem, Indag. Math. 34 (1972) 257--262. |
[7] | E. van Dam and W. Haemers, Graphs with constant μ and μ, Discrete Math. 182 (1998) 293--307, doi: 10.1016/S0012-365X(97)00150-7. |
[8] | H. Wilf, The friendship theorem in combinatorial mathematics and its applications, Proc. Conf. Oxford, 1969 (Academic Press: London and New York, 1971) 307--309. |
Received 21 May 2010
Revised 1 April 2011
Accepted 1 April 2011
Close