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- W2964343298 abstract "Let hom( G ) denote the size of the largest clique or independent set of a graph G . In 2007, Bukh and Sudakov proved that every n -vertex graph G with hom( G ) = O (log n ) contains an induced subgraph with Ω( n 1/2 ) distinct degrees, and raised the question of deciding whether an analogous result holds for every n -vertex graph G with hom( G ) = O ( n ϵ ), where ϵ > 0 is a fixed constant. Here, we answer their question in the affirmative and show that every graph G on n vertices contains an induced subgraph with Ω(( n /hom( G )) 1/2 ) distinct degrees. We also prove a stronger result for graphs with large cliques or independent sets and show, for any fixed k ∈ ℕ, that if an n -vertex graph G contains no induced subgraph with k distinct degrees, then hom( G )⩾ n /( k − 1) − o ( n ); this bound is essentially best possible." @default.
- W2964343298 created "2019-07-30" @default.
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- W2964343298 date "2017-08-01" @default.
- W2964343298 modified "2023-09-25" @default.
- W2964343298 title "Induced Subgraphs With Many Distinct Degrees" @default.
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- W2964343298 doi "https://doi.org/10.1017/s0963548317000256" @default.
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