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Contractible Edges in 3-Connected Cubic Graphs

    An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. Let EC(G) be the set of contractible edges of G, NG() be the set of vertices adjacent to three vertices of a triangle △. It has been proved that |EC(G)||V(G)|2 in a 3-connected graph G of order at least 5. In this note G is a 3-connected cubic graph containing k(k>0) triangles, at least 2k+9 vertices and with every NG() an independent set. Then |EC(G)||V(G)|2k+92. This is a bound better than |V(G)|2 under some conditions.

    Communicated by Eddie Cheng

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