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Minimum Clique Number, Chromatic Number, and Ramsey Numbers
Let Q(n, c) denote the minimum clique number over graphs with n vertices and chromatic number c. We investigate the asymptotics of Q(n, c) when n/c is held constant. We show that when n/c is an integer α, Q(n, c) has the same growth order as the inverse function of the Ramsey number R(α+ 1, t) (as a function of t). Furthermore, we show that if certain asymptotic properties of the Ramsey numbers...
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Let (G) be the minimum degree of a graph G: A number of results about trianglefree graphs determine the maximum chromatic number of graphs of order n with (G) n=3: In this paper these results are extended to Kr+1-free graphs of order n with (G) (1 2= (2r 1))n: In particular: (a) there exist Kr+1-free graphs of order n with (G) > (1 2= (2r 1))n o (n) and arbitrary large chromatic number; (b) if ...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 2019
ISSN: 0095-8956
DOI: 10.1016/j.jctb.2019.02.007