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DISTANCE-UNIFORM GRAPHS WITH LARGE DIAMETER.
- Source :
-
SIAM Journal on Discrete Mathematics . 2019, Vol. 33 Issue 2, p994-1005. 12p. - Publication Year :
- 2019
-
Abstract
- An ∊-distance-uniform graph is one with a critical distance d such that from every vertex, all but at most an ∊-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 ∊-distance-uniform graph (and, in fact, a broader class of ∊-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 1/n ≤ ∊ ≤ 1/logn, we construct ∊-distance-uniform graphs with critical distance 2Ω(logn/loge-1) 2O(logn/log e-1). We also prove an upper bound on the critical distance of the form 2Ologn/loge-1 for all e and n. Our lower bound construction introduces a novel method inspired by the Tower of Hanoi puzzle and may itself be of independent interest. [ABSTRACT FROM AUTHOR]
- Subjects :
- *GRAPH theory
*DIAMETER
Subjects
Details
- Language :
- English
- ISSN :
- 08954801
- Volume :
- 33
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- SIAM Journal on Discrete Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 139068292
- Full Text :
- https://doi.org/10.1137/17M115791X