Geodesic
On the intersection number of a~graph
Diskretnaya Matematika, Tome 19 (2007) no. 4, pp. 97-107.

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We find an expression of the intersection number of a graph in terms of the minimum number of complete subgraphs that form a covering of the graph. This provides us with a uniform approach to studying properties of the intersection number of a graph. We distinguish the class of graphs for which the intersection number is equal to the least number of cliques covering the graph. It is proved that the intersection number of a complete $r$-partite graph $r\overline K_2$ is equal to the least $n$ such that $r\le\binom{n-1}{[n/2]-1}$. It is proved that the intersection number of the graph $r\overline K_2+K_m$ is equal to the least $n$ such that $m+r\le2^{n-1}$, $r\le\binom{n-1}{[n/2]-1}$. Formulas for the intersection numbers of the graphs $rC_4$, $r\operatorname{Chain}(3)$, $r(C_4+K_m)$, $rW_5$ are obtained. 
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E. E. Marenich; N. S. Bol'shakova. On the intersection number of a~graph. Diskretnaya Matematika, Tome 19 (2007) no. 4, pp. 97-107. http://geodesic.mathdoc.fr/item/DM_2007_19_4_a5/

[1] Kharari F., Teoriya grafov, Mir, Moskva, 1973 | MR

[2] Szpilrajn-Marczewski E., “Sur deux propriétés des classes d'ensembles”, Fundam. Math., 33 (1945), 303–307 | MR | Zbl

[3] Erdős P., Goodman A. W., Pósa L., “The representation of a graph by set intersections”, Canad. J. Math., 18:1 (1966), 106–112 | MR | Zbl

[4] Scheinerman E. R., Trenk A. N., “On the fractional intersection number of graph”, Graph Comb., 15:3 (1999), 341–351 | DOI | MR | Zbl

[5] Marenich E. E., “Chislo peresechenii grafa”, Materialy VIII Mezhdunarodnogo seminara “Diskretnaya matematika i ee prilozheniya”, MGU, Moskva, 2004, 216–219

[6] Bollobás B., Erdős P., Spencer J., West D. B., “Clique covering of the edges of a random graph”, Combinatorica, 13 (1993), 1–5 | DOI | MR | Zbl

[7] Frieze A., Reed B., “Covering the edges of a random graph by cliques”, Combinatorica, 15 (1995), 489–497 | DOI | MR | Zbl

[8] Schönheim J., “On a problem of Purdy related to Sperner systems”, Canad. Math. Bull., 17 (1974), 135–136 | MR | Zbl