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- W2963664551 abstract "An $epsilon$-distance-uniform graph is one with a critical distance $d$ such that from every vertex, all but at most an $epsilon$-fraction of the remaining vertices are at distance exactly $d$. Motivated by the theory of network creation games, Alon, Demaine, Hajiaghayi, and Leighton made the following conjecture of independent interest: that every $epsilon$-distance-uniform graph (and, in fact, a broader class of $epsilon$-distance-almost-uniform graphs) has critical distance at most logarithmic in the number of vertices $n$. We disprove this conjecture and characterize the asymptotics of this extremal problem. Specifically, for $frac1n le epsilon le frac1{log n}$, we construct $epsilon$-distance-uniform graphs with critical distance $2^{Omega(frac{log n}{log epsilon^{-1}})}$. We also prove an upper bound on the critical distance of the form $2^{O(frac{log n}{log epsilon^{-1}})}$ for all $epsilon$ and $n$. Our lower bound construction introduces a novel method inspired by the Tower of Hanoi puzzle and may itself be of independent interest." @default.
- W2963664551 created "2019-07-30" @default.
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- W2963664551 date "2019-01-01" @default.
- W2963664551 modified "2023-09-24" @default.
- W2963664551 title "Distance-Uniform Graphs with Large Diameter" @default.
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- W2963664551 doi "https://doi.org/10.1137/17m115791x" @default.
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