Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

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 $U$. For the family of all undirected graphs on $n$ vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an induced universal graph with $O\!\left(2^{n/2}\right)$ vertices, matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965]. Let $k= \lceil D/2 \rceil$. Improving asymptotically on previous results by Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL 2008], we give an induced universal graph with $O\!\left(\frac{k2^k}{k!}n^k \right)$ vertices for the family of graphs with $n$ vertices of maximum degree $D$. For constant $D$, Butler gives a lower bound of $\Omega\!\left(n^{D/2}\right)$. For an odd constant $D\geq 3$, Esperet et al. and Alon and Capalbo [SODA 2008] give a graph with $O\!\left(n^{k-\frac{1}{D}}\right)$ vertices. Using their techniques for any (including constant) even values of $D$ gives asymptotically worse bounds than we present. For large $D$, i.e. when $D = \Omega\left(\log^3 n\right)$, the previous best upper bound was ${n\choose\lceil D/2\rceil} n^{O(1)}$ due to Adjiashvili and Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is ${\lfloor n/2\rfloor\choose\lfloor D/2 \rfloor}2^{\pm\tilde{O}\left(\sqrt{D}\right)}$. Hence the optimal size is $2^{\tilde{O}(D)}$ and our construction is within a factor of $2^{\tilde{O}\left(\sqrt{D}\right)}$ from this. The previous results were larger by at least a factor of $2^{\Omega(D)}$. As a part of the above, proving a conjecture by Esperet et al., we construct an induced universal graph with $2n-1$ vertices for the family of graphs with max degree $2$. In addition, we give results for acyclic graphs with max degree $2$ and cycle graphs. Our results imply the first labeling schemes that for any $D$ are at most $o(n)$ bits from optimal.

[1]  Brendan D. McKay,et al.  Asymptotic enumeration by degree sequence of graphs with degreeso(n1/2) , 1991, Comb..

[2]  Arnold L. Rosenberg,et al.  Universal Graphs for Bounded-Degree Trees and Planar Graphs , 1989, SIAM J. Discret. Math..

[3]  Ronald L. Graham,et al.  ON UNIVERSAL GRAPHS , 1979 .

[4]  Fan Chung Graham,et al.  On graphs which contain all small trees , 1978, J. Comb. Theory, Ser. B.

[5]  John Harold Muller,et al.  Local structure in graph classes , 1988 .

[6]  Amos Korman,et al.  Labeling schemes for vertex connectivity , 2007, TALG.

[7]  Noga Alon,et al.  Optimal induced universal graphs for bounded-degree graphs , 2016, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  Edith Cohen,et al.  Labeling dynamic XML trees , 2002, SIAM J. Comput..

[9]  Pierre Fraigniaud,et al.  Compact ancestry labeling schemes for XML trees , 2010, SODA '10.

[10]  J. Moon,et al.  On minimal n-universal graphs , 1965, Proceedings of the Glasgow Mathematical Association.

[11]  David Peleg,et al.  Informative labeling schemes for graphs , 2000, Theor. Comput. Sci..

[12]  Nicolas Bonichon,et al.  Short Labels by Traversal and Jumping , 2006, Electron. Notes Discret. Math..

[13]  Haim Kaplan,et al.  A comparison of labeling schemes for ancestor queries , 2002, SODA '02.

[14]  R. Graham,et al.  On graphs which contain all sparse graphs , 1982 .

[15]  Noga Alon,et al.  Optimal universal graphs with deterministic embedding , 2008, SODA '08.

[16]  David Peleg,et al.  Labeling Schemes for Weighted Dynamic Trees , 2003, ICALP.

[17]  Mikkel Thorup,et al.  Adjacency Labeling Schemes and Induced-Universal Graphs , 2014, STOC.

[18]  Ronald L. Graham,et al.  On Universal Graphs for Spanning Trees , 1983 .

[19]  Anita Liebenau,et al.  Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph , 2017, Journal of the European Mathematical Society.

[20]  Lenore Cowen,et al.  Compact routing with minimum stretch , 1999, SODA '99.

[21]  Pierre Fraigniaud,et al.  Routing in Trees , 2001, ICALP.

[22]  Arnaud Labourel,et al.  On induced-universal graphs for the class of bounded-degree graphs , 2008, Inf. Process. Lett..

[23]  Stephen Alstrup,et al.  Small induced-universal graphs and compact implicit graph representations , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[24]  Nicola Santoro,et al.  Labelling and Implicit Routing in Networks , 1985, Computer/law journal.

[25]  Gábor Rudolf,et al.  Minimal Universal Bipartite Graphs , 2007, Ars Comb..

[26]  Melvin A. Breuer,et al.  An unexpected result in coding the vertices of a graph , 1967 .

[27]  David Peleg,et al.  Compact and localized distributed data structures , 2003, Distributed Computing.

[28]  David Adjiashvili,et al.  Labeling Schemes for Bounded Degree Graphs , 2014, ICALP.

[29]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[30]  Noga Alon,et al.  Universality, Tolerance, Chaos and Order , 2010 .

[31]  Brendan D. McKay,et al.  Asymptotic Enumeration by Degree Sequence of Graphs of High Degree , 1990, Eur. J. Comb..

[32]  R. Rado Universal graphs and universal functions , 1964 .

[33]  David Peleg,et al.  Compact routing schemes with low stretch factor , 2003, J. Algorithms.

[34]  Moni Naor,et al.  Implicit representation of graphs , 1992, STOC '88.

[35]  P. Erdos-L Lovász Problems and Results on 3-chromatic Hypergraphs and Some Related Questions , 2022 .

[36]  Mikkel Thorup,et al.  Approximate distance oracles , 2001, JACM.

[37]  Fan Chung Graham,et al.  Universal graphs and induced-universal graphs , 1990, J. Graph Theory.

[38]  Gary Chartrand,et al.  The point-arboricity of a graph , 1968 .

[39]  Noga Alon,et al.  Asymptotically optimal induced universal graphs , 2017 .

[40]  Cyril Gavoille,et al.  Shorter Implicit Representation for Planar Graphs and Bounded Treewidth Graphs , 2007, ESA.

[41]  M. Chiani Error Detecting and Error Correcting Codes , 2012 .

[42]  Melvin A. Breuer,et al.  Coding the vertexes of a graph , 1966, IEEE Trans. Inf. Theory.

[43]  Pierre Fraigniaud,et al.  On randomized representations of graphs using short labels , 2009, SPAA '09.

[44]  J. Moon Topics on tournaments , 1968 .

[45]  Fan Chung,et al.  On the Decomposition of Graphs , 1981 .

[46]  Steve Butler,et al.  Induced-Universal Graphs for Graphs with Bounded Maximum Degree , 2009, Graphs Comb..

[47]  Haim Kaplan,et al.  Compact labeling schemes for ancestor queries , 2001, SODA '01.

[48]  Noga Alon,et al.  Sparse universal graphs for bounded‐degree graphs , 2007, Random Struct. Algorithms.