Independence in connected graphs
نویسندگان
چکیده
منابع مشابه
Independence in connected graphs
We prove that if G = (VG, EG) is a finite, simple, and undirected graph with κ components and independence number α(G), then there exist a positive integer k ∈ N and a function f : VG → N0 with non-negative integer values such that f(u) ≤ dG(u) for u ∈ VG, α(G) ≥ k ≥ ∑ u∈VG 1 dG(u)+1−f(u) , and ∑ u∈VG f(u) ≥ 2(k − κ). This result is a best-possible improvement of a result due to Harant and Schi...
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The Chvátal–Erdős Theorem states that every graph whose connectivity is at least its independence number has a spanning cycle. In 1976, Fouquet and Jolivet conjectured an extension: If G is an n-vertex k-connected graph with independence number a, and a ≥ k, then G has a cycle with length at least k(n+a−k) a . We prove this conjecture.
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For a set D of positive integers, we define a vertex set S ⊆ V (G) to be D-independent if u, v ∈ S implies the distance d(u, v) 6∈ D. The D-independence number βD(G) is the maximum cardinality of a D-independent set. In particular, the independence number β(G) = β{1}(G). Along with general results we consider, in particular, the odd-independence number βODD(G) where ODD = {1, 3, 5, . . .}.
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ژورنال
عنوان ژورنال: Discrete Applied Mathematics
سال: 2011
ISSN: 0166-218X
DOI: 10.1016/j.dam.2010.08.029