Independence in Direct-Product Graphs.

Ars Comb 01/1998; 50.
Source: DBLP
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    Electr. J. Comb. 01/2011; 18.
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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
    Acta Mathematica Sinica 01/2012; 28(4):697-706. · 0.48 Impact Factor
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    ABSTRACT: It was first shown by Cameron and Ku that the group $G=Sym(n)$ has the strict EKR property. Then Godsil and Meagher presented an entirely different proof of this fact using some algebraic properties of the symmetric group. A similar method was employed to prove that the projective general linear group $PGL(2,q)$, with its natural action on the projective line $\mathbb{P}_q$, has the strict EKR property. The main objective in this thesis is to formally introduce this method, which we call the module method, and show that this provides a standard way to prove Erdos-Ko-Rado theorems for other permutation groups. We then, along with proving Erdos-Ko-Rado theorems for various groups, use this method to prove some permutation groups have the strict EKR property. We will also show that this method can be useful in characterizing the maximum independent sets of some Cayley graphs. To explain the module method, we need some facts from representation theory of groups, in particular, the symmetric group. We will provide the reader with a sufficient level of background from representation theory as well as graph theory and linear algebraic facts about graphs.


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