## On super connectivity of Cartesian product graphs.(English)Zbl 1153.05034

The super connectivity $$\kappa_1$$, a more refined parameter than the connectivity $$\kappa$$, of a connected graph $$G$$ is the minimum number of vertices whose deletion results in a disconnected graph without isolated vertices.
This article provides bounds for $$\kappa_1$$ of the Cartesian product of two connected graphs, that is, $\min\{\kappa(G_1)+2\kappa(G_2), 2\kappa(G_1)+\kappa(G_2)\}\leq\kappa_1(G_1\times G_2)\leq\min\{m\kappa(G_2), n\kappa(G_1)\}$ if $$G_1\neq K_m$$ and $$G_2\neq K_n$$, which generalizes the result of B. S. Shieh [“Super edge- and point-connectivities of the Cartesian product of regular graphs”, Networks 40, No. 2, 91–96 (2002; Zbl 1018.05056)] on the super connectedness of the Cartesian product of two regular graphs with maximum connectivity.
Particularly, this paper determines that $$\kappa_1(K_m\times K_n)=\min\{m+2n-4, 2m+n-4\}$$ for $$m+n\geq 6$$ and states sufficient conditions to guarantee $$\kappa_1(K_2\times G)=2\kappa(G)$$. As a consequence, it is immediately obtained that the super connectivity of the $$n$$-cube is $$2n-2$$ for $$n\geq 3$$.

### MSC:

 05C40 Connectivity 05C35 Extremal problems in graph theory

Zbl 1018.05056
Full Text:

### References:

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