Document Type

Article

Publication Date

10-1-2025

Abstract

For integers (Formula presented.), a graph (Formula presented.) is (Formula presented.) -stable if (Formula presented.) for every (Formula presented.) with (Formula presented.). A recent result of Dong and Wu states that every (Formula presented.) -stable graph (Formula presented.) satisfies (Formula presented.). A (Formula presented.) -stable graph (Formula presented.) is tight if (Formula presented.); and (Formula presented.) -tight for some integer (Formula presented.) if (Formula presented.). In this paper, we first prove that for all (Formula presented.), the only tight (Formula presented.) -stable graphs are (Formula presented.) and (Formula presented.), answering a question of Dong and Luo. We then prove that for all nonnegative integers (Formula presented.) with (Formula presented.), every (Formula presented.) -tight (Formula presented.) -stable graph has at most (Formula presented.) vertices, answering a question of Dong and Luo in the negative.

Publication Source (Journal or Book title)

Journal of Graph Theory

First Page

193

Last Page

199

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