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Large induced trees in sparse random graphs

โœ Scribed by A.M Frieze; B Jackson


Publisher
Elsevier Science
Year
1987
Tongue
English
Weight
621 KB
Volume
42
Category
Article
ISSN
0095-8956

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A study of the orders of maximal induced trees in a random graph G, with small edge probability p is given. In particular, it is shown that the giant component of almost every G,, where p = c/n and c > 1 is a constant, contains only very small maximal trees (that are of a specific type) and very lar

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The author proved that, for c > 1, the random graph G(n, p ) on n vertices with edge probability p = c / n contains almost always an induced tree on at least q n ( 1 -o( 1)) vertices, where L Y ~ is the positive root of the equation CLY = log( 1 + c'a). It is shown here that if c is sufficiently lar

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## Abstract For a graph __G__, let __a__(__G__) denote the maximum size of a subset of vertices that induces a forest. Suppose that __G__ is connected with __n__ vertices, __e__ edges, and maximum degree ฮ”. Our results include: (a) if ฮ”โ€‰โ‰คโ€‰3, and __G__โ€‰โ‰ โ€‰__K__~4~, then __a__(__G__)โ€‰โ‰ฅโ€‰__n__โ€‰โˆ’โ€‰e/4โ€‰โˆ’โ€‰1

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The performance of the greedy coloring algorithm ''first fit'' on sparse random graphs G and on random trees is investigated. In each case, approximately n, c r n log log n colors are used, the exact number being concentrated almost surely on at 2 most two consecutive integers for a sparse random gr