Near-Optimal Induced Universal Graphs for Bounded Degree Graphs
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Near-Optimal Induced Universal Graphs for Bounded Degree Graphs. / Abrahamsen, Mikkel; Alstrup, Stephen; Holm, Jacob; Knudsen, Mathias Bæk Tejs; Stöckel, Morten.
44th International Colloquium on Automata, Languages, and Programming (ICALP 201. ed. / Ioannis Chatzigiannaki; Piotr Indyk; Fabian Kuhn; Anca Muscholl. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. p. 1-14 128 (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 80).Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review
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TY - GEN
T1 - Near-Optimal Induced Universal Graphs for Bounded Degree Graphs
AU - Abrahamsen, Mikkel
AU - Alstrup, Stephen
AU - Holm, Jacob
AU - Knudsen, Mathias Bæk Tejs
AU - Stöckel, Morten
N1 - Conference code: 44
PY - 2017
Y1 - 2017
N2 - A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U. We give upper and lower bounds for the size of induced universal graphs for the family of graphs with n vertices of maximum degree D. Our new bounds improve several previous results except for the special cases where D is either near-constant or almost n/2. For constant even D Butler [Graphs and Combinatorics 2009] has shown O(n^(D/2)) and recently Alon and Nenadov [SODA 2017] showed the same bound for constant odd D. For constant D Butler also gave a matching lower bound. For generals graphs, which corresponds to D = n, Alon [Geometric and Functional Analysis, to appear] proved the existence of an induced universal graph with (1+o(1)) \cdot 2^((n-1)/2) vertices, leading to a smaller constant than in the previously best known bound of 16 * 2^(n/2) by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015]. In this paper we give the following lower and upper bound of binom(floor(n/2))(floor(D/2)) * n^(-O(1)) and binom(floor(n/2))(floor(D/2)) * 2^(O(sqrt(D log D) * log(n/D))), respectively, where the upper bound is the main contribution. The proof that it is an induced universal graph relies on a randomized argument. We also give a deterministic upper bound of O(n^k / (k-1)!). These upper bounds are the best known when D <= n/2 - tilde-Omega(n^(3/4)) and either D is even and D = omega(1) or D is odd and D = omega(log n/log log n). In this range we improve asymptotically on the previous best known results by Butler [Graphs and Combinatorics 2009], Esperet, Arnaud and Ochem [IPL 2008], Adjiashvili and Rotbart [ICALP 2014], Alon and Nenadov [SODA 2017], and Alon [Geometric and Functional Analysis, to appear].
AB - A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U. We give upper and lower bounds for the size of induced universal graphs for the family of graphs with n vertices of maximum degree D. Our new bounds improve several previous results except for the special cases where D is either near-constant or almost n/2. For constant even D Butler [Graphs and Combinatorics 2009] has shown O(n^(D/2)) and recently Alon and Nenadov [SODA 2017] showed the same bound for constant odd D. For constant D Butler also gave a matching lower bound. For generals graphs, which corresponds to D = n, Alon [Geometric and Functional Analysis, to appear] proved the existence of an induced universal graph with (1+o(1)) \cdot 2^((n-1)/2) vertices, leading to a smaller constant than in the previously best known bound of 16 * 2^(n/2) by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015]. In this paper we give the following lower and upper bound of binom(floor(n/2))(floor(D/2)) * n^(-O(1)) and binom(floor(n/2))(floor(D/2)) * 2^(O(sqrt(D log D) * log(n/D))), respectively, where the upper bound is the main contribution. The proof that it is an induced universal graph relies on a randomized argument. We also give a deterministic upper bound of O(n^k / (k-1)!). These upper bounds are the best known when D <= n/2 - tilde-Omega(n^(3/4)) and either D is even and D = omega(1) or D is odd and D = omega(log n/log log n). In this range we improve asymptotically on the previous best known results by Butler [Graphs and Combinatorics 2009], Esperet, Arnaud and Ochem [IPL 2008], Adjiashvili and Rotbart [ICALP 2014], Alon and Nenadov [SODA 2017], and Alon [Geometric and Functional Analysis, to appear].
U2 - 10.4230/LIPIcs.ICALP.2017.128
DO - 10.4230/LIPIcs.ICALP.2017.128
M3 - Article in proceedings
SN - 978-3-95977-041-5
T3 - Leibniz International Proceedings in Informatics (LIPIcs)
SP - 1
EP - 14
BT - 44th International Colloquium on Automata, Languages, and Programming (ICALP 201
A2 - Chatzigiannaki, Ioannis
A2 - Indyk, Piotr
A2 - Kuhn, Fabian
A2 - Muscholl, Anca
PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik
Y2 - 10 July 2017 through 14 July 2017
ER -
ID: 196763418