EMBEDDING THE COMPLEMENT OF A COMPLETE GRAPH IN A FINITE PROJECTIVE PLANE

Yıl 2015, Cilt: 3 Sayı: 1, 121 - 125, 01.04.2015

İbrahim Günaltı

Öz

Let S = (P;L) be a non-trivial regular nite linear space with v points, v + k lines, k  3: We show that if S contains at least 􀀀k 2
lines of size b(p) 􀀀 2 and one line size b(p) for some point p, then S is embeddable in a unique projective plane  of order b(p) 􀀀 1 and  􀀀 s is a complete graph of order k ; where b(p)  4 for some point p:

Anahtar Kelimeler

linear space, projective plane, complete graph

Kaynakça

• [1] Batten , L.M. and Beutelspacher, A. ; Combinatorics of points and lines, Cambridge Univer- sity Press, 1993.
• [2] Batten, L.M. ; Embedding pseudo-complements in nite projective planes, Ars Combin. 24 (1987), 129-132.
• [3] Bose , R.C. and Shrikhande, S.S. ; Embedding the complement of a oval in a projective plane of even order, Discrete Math. 6 (1973), 305-312.
• [4] Bruck, R. H. ; Existence problems for classes of nite projective planes, Lectures delivered to the Canadian Math. Congress, Sask., Aug.1963.
• [5] De Brujin N.G and Erdos, P. ; On a combinatorial problem, Nederl Akad.Wetemsch. proc. Sect. Sci. 51 (1948), 1277 - 1279.
• [6] De Witte, P. ;The exceptional case in a Theorem of Bose and Shrikhande, J. Austral.Math. soc. 24 (Series A) (1977), 64-78.
• [7] Dickey, L. J. ; Embedding the complement of a unital in a projective plane, Atti del convegno di Geometria Combinatoria e sue Applicazioni, Perugia, 1971, pp. 199-203.
• [8] Gunaltl, _I. and Olgun, S. ; On the embedding some linear spaces in nite projective planes. J.geom. 68 (2000) 96-99.
• [9] Gunaltl, _I. , Anapa, P. and Olgun, S. ; On the embedding of complements of some hyperbolic planes. Ars Combin. 80 (2006), pp. 205-214.
• [10] Hall, M. ; Projective planes, Trans. Amer. Math. Soc. 54 (1943) 229-277.
• [11] Kaya, R. and  Ozcan, E. ; On the construction of B-L planes from projective planes, Rendiconti del Seminario Matematico Di Bresciot (1984), pp. 427-434.
• [12] Mullin, R.C. and Vanstone, S.A. ; Embedding the pseudo-complements of a quadrilateral in a nite projective plane, Ann.New York Acad.Sci.319, 405-412.
• [13] Totten, J. ; Embedding the complement of two lines in a nite projective plane, J.Austral.Math.Soc. 22 (Series A) (1976), 27-34.