On edge connectivity of direct products of graphs |
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Authors: | Xiang-Lan Cao Elkin Vumar |
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Affiliation: | a College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, PR China b University of Maribor, FME, Smetanova 17, 2000 Maribor, Slovenia |
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Abstract: | Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u1,v1) and (u2,v2) are adjacent in G×H if u1u2∈E(G) and v1v2∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(2(n/logn)). |
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Keywords: | Combinatorial problems Connectivity Direct product Graph product Separating set |
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