Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

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 - 2017/7/1

Y1 - 2017/7/1

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 (nD/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)) · 2(n-1)/2 vertices, leading to a smaller constant than in the previously best known bound of 16 · 2n/2 by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015]. In this paper we give the following lower and upper bound of (⌊n/2⌋ ⌊D/2⌋ · n-O(1) and (⌊n/2⌋ ⌊D/2⌋ · 2O(√DlogD·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 (nκ/(κ-1)!). These upper bounds are the best known when D ≤ n/2 -Ω(n3/4) and either D is even and D = ω(1) or D is odd and D = ω(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 (nD/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)) · 2(n-1)/2 vertices, leading to a smaller constant than in the previously best known bound of 16 · 2n/2 by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015]. In this paper we give the following lower and upper bound of (⌊n/2⌋ ⌊D/2⌋ · n-O(1) and (⌊n/2⌋ ⌊D/2⌋ · 2O(√DlogD·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 (nκ/(κ-1)!). These upper bounds are the best known when D ≤ n/2 -Ω(n3/4) and either D is even and D = ω(1) or D is odd and D = ω(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

T2 - 44th International Colloquium on Automata, Languages, and Programming

Y2 - 10 July 2017 through 14 July 2017

ER -