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For a positive integer r and a vertex v of a graph G , let I G ( r ) ( v ) denote the set of independent sets of G that have exactly r elements and contain v . Motivated by a problem of Holroyd and Talbot, Hurlbert and Kamat conjectured that for any r and any tree T , there exists a leaf z of T such that | I T ( r ) ( v ) | ≤ | I T ( r ) ( z...
Let σ 4 ∗ = min{ ∑ i = 1 4 d ( v i ) + | ⋃ i = 1 4 N ( v i ) | − | ⋂ i = 1 4 N ( v i ) | : { v 1 , v 2 , v 3 , v 4 } is an independent set of a graph G}. In this paper, we give a low bound for the length of a longest cycle in a 4-connected graph and get the following result: If G is a 4-connected graph on n vertices,...
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