
TL;DR
This paper disproves a conjecture about the maximum number of independent sets containing a specific vertex in trees, showing counterexamples for larger values of r where the conjecture fails.
Contribution
The authors construct explicit counterexamples to the conjecture for r ≥ 5, demonstrating the conjecture's limitations in certain cases.
Findings
Counterexamples for r ≥ 5 in trees.
The conjecture does not hold universally for all r.
Largest r with non-empty independent sets at x is 2k+1.
Abstract
For a positive integer and a vertex of a graph , let denote the set of all independent sets of that have exactly elements and contain . Hurlbert and Kamat conjectured that for any and any tree , there exists a leaf of such that for each vertex of . They proved the conjecture for . For any , we construct a tree that has a vertex such that is not a leaf of , for any leaf of and any , and is the largest integer for which is non-empty. Therefore, the conjecture is not true for .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
