Complete tripartite subgraphs in the coprime graph of integers
نویسندگان
چکیده
منابع مشابه
Complete tripartite subgraphs in the coprime graph of integers
We denote by f(n, k) the number of positive integers m no,AC{1,2,...,n} with lAl>f(n,2) (if61n then f(n,2)= in), then the coprime graph induced by A contains a complete tripartite graph on 2 Lc,,,~~~;,,, J + 1 vertices where one of the classes ...
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In this paper we study cycles in the coprime graph of integers. We denote by f(n, k) the number of positive integers m ≤ n with a prime factor among the first k primes. We show that there exists a constant c such that if A ⊂ {1, 2, . . . , n} with |A| > f(n, 2) (if 6|n then f(n, 2) = 2 3 n), then the coprime graph induced by A not only contains a triangle, but also a cycle of length 2l + 1 for ...
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Abstract Let (V,E) be the coprime graph with vertex set V = {1, 2, . . . , n} and edges (i, j) ∈ E if gcd(i, j) = 1. We determine the kernels of the coprime graph and its loopless counterpart as well as so-called simple bases for them (in case such bases exist), which means that basis vectors have entries only from {−1, 0, 1}. For the loopless version knowledge about the value distribution of M...
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d|en d denote the number and the sum of exponential divisors of n, respectively. Properties of these functions were investigated by several authors, see [1], [2], [3], [5], [6], [8]. Two integers n,m > 1 have common exponential divisors iff they have the same prime factors and for n = ∏r i=1 p ai i , m = ∏r i=1 p bi i , ai, bi ≥ 1 (1 ≤ i ≤ r), the greatest common exponential divisor of n and m is
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1999
ISSN: 0012-365X
DOI: 10.1016/s0012-365x(98)00359-8