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Floor function proof

Assume that G does not equal K_2.

K_2 is the complete 2-partite graph

If G is regular of degree "r" and Kappa(G) = 1, prove that &#955;(G) <= [floor function of "r / 2" ]

The floor function, also called the greatest integer function or integer value, gives the largest integer less than or equal to "r / 2."

DEFN: the connectivity, Kappa(G), of a graph G is the minimum number of vertices whose removal makes G disconnected, or trival.

DEFN: the edge connectivity, &#955;(G), is the minimum number of edges whose removal makes G disconnected or trival.

Solution Preview

Without loss of generality, we can assume that the graph is connected. Now, K_2 is the only 1-regular graph (graph of regularity 1). Hence we have G r-regular with r >= 2.

Consider any graph such that Kappa(G) = 1. By definition of Kappa, there ...

Solution Summary

This is a proof regarding a floor function.