### INCLUSION AND EXCLUSION FOR FINITELY MANY TYPES OF PROPERTIES

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Chae, Gab-Byoung;Cheong, Min-Seok;Kim, Sang-Mok

• 투고 : 2009.09.09
• 심사 : 2010.03.11
• 발행 : 2010.03.25
• 11 2

#### 초록

Inclusion and exclusion is used in many papers to count certain objects exactly or asymptotically. Also it is used to derive the Bonferroni inequalities in probabilistic area [6]. Inclusion and exclusion on finitely many types of properties is first used in R. Meyer [7] in probability form and first used in the paper of McKay, Palmer, Read and Robinson [8] as a form of counting version of inclusion and exclusion on two types of properties. In this paper, we provide a proof for inclusion and exclusion on finitely many types of properties in counting version. As an example, the asymptotic number of general cubic graphs via inclusion and exclusion formula is given for this generalization.

#### 키워드

inclusion and exclusion;asymptotic enumeration

#### 참고문헌

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2. Bela Bollabas, A probabilistic proof of an asymptotic formula for the number of labeled regular graphs, Europ. J. Combinatories 1 (1980), 311-316. https://doi.org/10.1016/S0195-6698(80)80030-8
3. G.-B. Chae, Asymptotic number of General Cubic Graphs with given connectivity, Journal of the Korean Math. Soc. 42, (2005), No.6, 1187-1203. https://doi.org/10.4134/JKMS.2005.42.6.1187
4. G.-B. Chae, Asymptotic number of General 4-Regular Graphs with given connectivity, Bulletin of the Korean Math. Soc. 43, (2006), No. 1, 125-140. https://doi.org/10.4134/BKMS.2006.43.1.125
5. G. Chartrand and L. Lesniak, Graphs and Digraphs, Fourth Edition, Chapman & Hall/CRC, Boca Raton (2005).
6. Janos Galambos and Italo Simonelli, Bonferroni-type Inequalities with applications, (Springer-Verlag, 1996).
7. R.M. Meyer, Note on a 'multivariate' form of Bonferroni's inequalities, The annals of Mathematical Statistics 40 2 (1969), 692-693. https://doi.org/10.1214/aoms/1177697743
8. B.D. McKay, E.M. Palmer, R. C. Read and R.W. Robinson, The asymptotic number of claw-free cubic graphs, Discrete Math. 272 (2003), 107-118. https://doi.org/10.1016/S0012-365X(03)00188-2
9. E.M. Palmer, Graphical Evolution, (John Wiley & Sons, 1985).
10. N.C. Wormald, Some problems in the enumeration of labelled graphs, Doctoral Thesis, (University of Newcastle, Callaghan, NSW, Australia, 1978).