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Hello everyone. I'm really struggling with this question. All help appreciated.

Find the minimum positive integer r for which there exists an r-regular graph G such that λ(G) ≥ κ(G) + 2

I know it's not 1,2,3-regular since κ(G) = λ(G) for those graphs.

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This is a homework question:… – domotorp Apr 9 '12 at 8:11
Actually it's an exercise from Chartrand & Lesniak Ex2.4 Q.16, and it's not homework. – janvdl Apr 9 '12 at 8:32
I'd suggest you put such question on . – Jernej Apr 9 '12 at 9:14
Just consider all the possibilities, there are very few. – Brendan McKay Apr 9 '12 at 14:56

Take tow disjoint copies of the complete graph on 5 vertices. Pick a vertex on each of the copy and split into tow vertices of degree 2. Match the the degree 2 vertices of both copies the identify them to get 4-regular, 4-egde-connected, and 2-vertex-connected graph.

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