Double-Wheel Graphs Are Graceful

We present the first polynomial time construction procedure for generating graceful double-wheel graphs. A graph is graceful if its vertices can be labeled with distinct integer values from {0;..., e}, where e is the number of edges, such that each edge has a unique value corresponding to the absolute difference of its endpoints. Graceful graphs have a range of practical application domains, including in radio astronomy, X-ray crystallography, cryptography, and experimental design. Various families of graphs have been proven to be graceful, while others have only been conjectured to be. In particular, it has been conjectured that so-called double-wheel graphs are graceful. A double-wheel graph consists of two cycles of N nodes connected to a common hub. We prove this conjecture by providing the first construction for graceful double-wheel graphs, for any N > 3, using a framework that combines streamlined constraint reasoning with insights from human computation. We also use this framework to provide a polynomial time construction for diagonally ordered magic squares.

[1]  Luis von Ahn,et al.  Human computation , 2009, 2009 46th ACM/IEEE Design Automation Conference.

[2]  Clifford A. Pickover.  The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across , 2002 .

[3]  Jean-François Puget,et al.  Program Does Not Equal Program: Constraint Programming and Its Relationship to Mathematical Programming , 2001, Interfaces.

[4]  Toby Walsh,et al.  Symmetry within Solutions , 2010, AAAI 2010.

[5]  Brigitte Jaumard,et al.  A Constraint-Based Approach to the Golomb Ruler Problem , 2007 .

[6]  G. Bloom.  A CHRONOLOGY OF THE RINGEL‐KOTZIG CONJECTURE AND THE CONTINUING QUEST TO CALL ALL TREES GRACEFUL * , 1979 .

[7]  Barbara M. Smith,et al.  Symmetry Breaking in Graceful Graphs , 2003, CP.

[8]  Barbara M. Smith.  Constraint Programming Models for Graceful Graphs , 2006, CP.

[9]  Joseph A. Gallian,et al.  A Dynamic Survey of Graph Labeling , 2009, The Electronic Journal of Combinatorics.

[10]  Roberto Frucht,et al.  GRACEFUL NUMBERING OF WHEELS AND RELATED GRAPHS , 1979 .

[11]  Jean-François Puget.  An Efficient Way of Breaking Value Symmetries , 2006, AAAI.

[12]  Timothy A. Redl.  Graceful Graphs and Graceful Labelings: Two Mathematical Programming Formulations and Some Other New Results , 2003 .

[13]  Barry O'Sullivan,et al.  Trends in Constraint Programming , 2007 .

[14]  Ronan Le Bras,et al.  Polynomial Time Construction for Spatially Balanced Latin Squares , 2012 .

[15]  Armando Tacchella,et al.  Theory and Applications of Satisfiability Testing , 2003, Lecture Notes in Computer Science.

[16]  Meinolf Sellmann,et al.  Streamlined Constraint Reasoning , 2004, CP.

[17]  Mirostaw Truszczynski.  GRACEFUL UNICYCLIC GRAPHS , 1984 .

[18]  Ronan Le Bras,et al.  From Streamlined Combinatorial Search to Efficient Constructive Procedures , 2012, AAAI.

[19]  Peter Jeavons,et al.  Symmetry Definitions for Constraint Satisfaction Problems , 2005, CP.

[20]  Peter Kulchyski.  and , 2015 .

[21]  Steve Linton,et al.  Conditional Symmetry Breaking , 2005, CP.

[22]  Michal Kouril,et al.  Resolution Tunnels for Improved SAT Solver Performance , 2005, SAT.