Combinatorial Mathematics V by C. H. C. Little

By C. H. C. Little

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Example text

As | P F I N A L | < n/(k + 1) always, the theorem follows. Remark. We can actually say more about /(c). For Ac small, / ( c + Ac) — /(c) ~ — (Ac)/(c) fc+1 as, roughly, an Eve starting at time c + Ac might have a birth in time interval [c, c + Ac), all of whose children survive, while Eve has no births in [0, c), all of whose children survive. Letting Ac —> 0 yields the differential equation f'(c) = —f(c)k+1. The initial valué /(O) = 1 gives a unique solution f(c) = (1 + ck)~llk. It is intriguing to plug in c = D.

But if it is still dangerous then check its second coin. If it is heads then change the color of d, otherwise do nothing. We cali the coloring at the time of termination the final coloring. We say the algorithm fails if some e e H is monochromatic in the final coloring. We shall bound the failure probability by fc(l — p)n + k2p. 1 then assures us that with positive probability the algorithm succeeds. This, by our usual magic, means that there is some running of the algorithm which yields a final coloring with no monochromatic e; that is, there exists a two-coloring of V with no monochromatic edge.

The approach seems most effective when dealing with random orderings. We give two examples. PropertyB. Wemodify the proof that m(n) = ü(2nn1/2ln~1^2 n) oftheprevious section. We assign to each vértex v £ V a "birth time" xv. The xv are independent real variables, each uniform in [0,1]. The ordering of V is then the ordering (under less than) of the xv. We now claim Pr[Bef}

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