Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  485.05052
Autor:  Erdös, Paul
Title:  On the covering of the vertices of a graph by cliques. (In English)
Source:  J. Math. Res. Expo. 1982, No.1, 93-96 (1982).
Review:  Let G(n) be a graph of n vertices. Denote by f(G(n)) = t the smallest integer for which the vertices of G(n) can be covered by t cliques. Denote furthr by h(G(n)) = \ell the largest integer for which there are \ell edges of G(n), no two are in the same clique. K. R. Parthasarathy and S.A.Choudum [J. Math. Phys. Sci. Madras 10, 255-261 (1976; Zbl 335.05125)] conjectured that if G(n) has no isolated vertices then (1) f(G(n)) \leq h(G(n)) holds for all graphs. A simple application of the probability method shows that (1) fails for almost graphs, as shown in the following theorem: There are positive absolute constants c1 and c2 for which for n > m0(c1,c2) c1\frac{n}{(log n)3} < max \frac{f(G(n))}{h(G(n))} < c2\frac{n}{(log n)3}.
Reviewer:  J.-H.Tian
Classif.:  * 05C70 Factorization, etc.
                   05C30 Enumeration of graphs and maps
                   05C35 Extremal problems (graph theory)
Keywords:  clique covering
Citations:  Zbl.335.05126

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