Finding Hamilton cycles in random graphs with few queries

We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of ${\mathcal G}(n,p)$ in order to typically find a subgraph possessing a given target property. We show that if $p\geq \frac{\ln n+\ln\ln n+\omega(1)}{n}$, then one can find a Hamilton cycle with high probability after exposing $(1+o(1))n$ edges. Our result is tight in both $p$ and the number of exposed edges.

[1]  Frank Harary,et al.  Graph Theory , 2016 .

[2]  Alan M. Frieze,et al.  Algorithmic theory of random graphs , 1997, Random Struct. Algorithms.

[3]  János Komlós,et al.  Limit distribution for the existence of Hamiltonian cycles in a random graph , 2006, Discret. Math..

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

[5]  N. Alon,et al.  The Probabilistic Method: Alon/Probabilistic , 2008 .

[6]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[7]  L. Pósa,et al.  Hamiltonian circuits in random graphs , 1976, Discret. Math..

[8]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[9]  Benny Sudakov,et al.  Finding paths in sparse random graphs requires many queries , 2015, Random Struct. Algorithms.

[10]  Alan M. Frieze,et al.  Hamilton cycles in random graphs and directed graphs , 2000, Random Struct. Algorithms.

[11]  Michael Krivelevich,et al.  Generating random graphs in biased Maker-Breaker games , 2015, Random Struct. Algorithms.

[12]  Benny Sudakov,et al.  Avoiding small subgraphs in Achlioptas processes , 2007, Random Struct. Algorithms.

[13]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[14]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[15]  Alan M. Frieze,et al.  Hamilton Cycles in a Class of Random Directed Graphs , 1994, J. Comb. Theory, Ser. B.

[16]  Benny Sudakov,et al.  The phase transition in random graphs: A simple proof , 2012, Random Struct. Algorithms.

[17]  Alan M. Frieze,et al.  Hamilton cycles in 3-out , 2009, Random Struct. Algorithms.