Independence in Direct-Product Graphs.

Ars Combinatoria -Waterloo then Winnipeg- (Impact Factor: 0.26). 01/1998; 50.
Source: DBLP

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    • "Received June 4, 2010, accepted November 24, 2010 The second author is supported by National Natural Foundation of China (Grant No. 10731040) and Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20093127110001); the third author is supported by National Natural Foundation of China (Grant No. 11001249) and Zhejiang Innovation Project (Grant No. T200905) 1) Corresponding author It is obvious that α(G × H) ≥ max{α(G)|H|, α(H)|G|}. Jha and Klavžar [1] "
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    ABSTRACT: Let Circ(r, n) be a circular graph. It is well known that its independence number α(Circ(r, n)) = r. In this paper we prove that a(Circ(r,n) H) = max{ r|H|,na(H)}\alpha (Circ(r,n) \times H) = \max \{ r|H|,n\alpha (H)\} for every vertex transitive graph H, and describe the structure of maximum independent sets in Circ(r, n) × H. As consequences, we prove a(G H) = max{ a(G)|V(H)|,a(H)|V(G)|}\alpha (G \times H) = \max \{ \alpha (G)|V(H)|,\alpha (H)|V(G)|\} for G being Kneser graphs, and the graphs defined by permutations and partial permutations, respectively. The structure of maximum independent sets in these direct products is also described. KeywordsVertex-transitivity–primitivity–independence number
    Preview · Article · Apr 2012 · Acta Mathematica Sinica
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    • "This lemma has many applications in extremal combinatorics and graph theory (see [6] [8] [9] [10] [11] [12] [14] [15] [16]). For B ⊂ V (G), let G[B] denote the sub-graph of G induced by B. Then, in Lemma 2.1, by taking G ′ as an induced subgraph G[B] and φ as the embedding mapping, we obtain the following lemma. "

    Preview · Article · Jan 2011 · The electronic journal of combinatorics
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    • "It is natural to ask whether the equality holds or not. In general, the equality does not hold for non-vertex-transitive graphs (see [13]). So Tardif [17] posed the following problem. "
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    ABSTRACT: The direct product $G\times H$ of graphs $G$ and $H$ is defined by: \[V(G\times H)=V(G)\times V(H)\] and \[E(G\times H)=\left\{[(u_1,v_1),(u_2,v_2)]: (u_1,u_2)\in E(G) \mbox{\ and\ } (v_1,v_2)\in E(H)\right\}.\] In this paper, we will prove that the equality $$\alpha(G\times H)=\max\{\alpha(G)|H|, \alpha(H)|G|\}$$ holds for all vertex-transitive graphs $G$ and $H$, which provides an affirmative answer to a problem posed by Tardif (Discrete Math. 185 (1998) 193-200). Furthermore, the structure of all maximum independent sets of $G\times H$ are determined. Comment: 11 pages
    Preview · Article · Jul 2010 · Journal of Combinatorial Theory Series B
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