Highly Connected Subgraphs of Graphs with Given Independence Number (Extended Abstract)

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Abstract

Let G be a graph on n vertices with independence number α. What is the largest k-connected subgraph that G must contain? We prove that if n is sufficiently large (nα2k+1 will do), then G contains a k-connected subgraph on at least n/α vertices. This is sharp, since G might be the disjoint union of α equally-sized cliques. For k3 and α=2,3, we shall prove that the same result holds for n4(k1) and n27(k1)4 sharp.

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Supported by JSPS KAKENHI, grant number 15K04979.

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