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%Optional fields: volume, series, address, edition, month, note, key 

%ARTICLE
%Required fields: author, title, journal, year
%Optional fields: volume, number, pages, month, note, key

%INPROCEEDINGS

%Required fields: author, title, booktitle, year
%Optional fields: editor, pages, organization, publisher, address, month, note, key 

@Article{alontarsi,
     author    = "N. Alon and M. Tarsi",
     title     = "Colorings and orientations of graphs",
     journal   = "Combinatorica",
     volume    = "12",
     year      = "1992",
     pages     = "125--134"
}

@Article{alonsurvey,
     author    = "N. Alon",
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     journal   = "Combinatorics, Prob. and Computing",
     volume    = "8",
     year      = "1999",
     pages     = "7--29"
}

@Article{intdom_ieq,
     author    = {B. Bre\v{s}ar and M. A. Henning and S. Klav\v{z}ar},
     title     = {ON INTEGER DOMINATION IN GRAPHS AND {V}IZING-LIKE PROBLEMS},
     journal   = {Taiwanese Journal of Mathematics},
     volume    = {10},
     number = {5},
     year      = {2006},
     pages     = {1317--1328}
}

@Misc{susan_ds,
   author = {K. Choudhary and S. Margulies and I.V. Hicks},
   title = {A {N}ote on {T}otal and {P}aired {D}omination of {C}artesian {P}roduct {G}raphs},
   howpublished ={Available at \/ {\tt http://arxiv.org/abs/1109.2174}},
}

@Misc{cite_cocoa,
   author = {J. Abbott and A.M. Bigatti},
   title = {{CoCoALib}: a {C}++ library for doing {C}omputations
in {C}ommutative {A}lgebra},
   howpublished ={Available at \/ {\tt http://cocoa.dima.unige.it/cocoalib}},
}


@Book{coxetal,
  author = 	 {D. Cox and J. Little and D. O'Shea},
  title = 	 {Ideals, Varieties and Algorithms},
  publisher = 	 {Springer},
  year = 	 {1998},
  address = 	 {New York}
}


@Book{coxetal2,
  author = 	 {D. Cox and J. Little and D. O'Shea},
  title = 	 {Using algebraic geometry},
  publisher = 	 {Springer},
  year = 	 {1997},
  address = 	 {New York},
  edition = 	 {Second}
}

@BOOK{diestel,
	AUTHOR = {R. Diestel},
	TITLE = {Graph Theory},
	PUBLISHER = {Springer-Verlag},
	YEAR = {2006},
	VOLUME = {173},
	SERIES = {Graduate Texts in Mathematics},
	EDITION = {3rd}
}

@INPROCEEDINGS{intdom1,
	TITLE = {Relationships between integer and fractional parameters of graphs},
	AUTHOR = {G. S. Domke and S. T. Hedetniemi and R. C. Laskar and G. H. Fricke},
	BOOKTITLE = {Graph Theory, Combinatorics, and Applications},
	ORGANIZATION = {Sixth Quadrennial International Conference on the Theory and Applications of Graphs},
	YEAR = {1991},
	PAGES = {371-387}
}



@BOOK{dummit,
	AUTHOR = {D.S. Dummit and R.M. Foote},
	TITLE = {Abstract algebra},
	PUBLISHER = {Wiley},
	YEAR = {1999}
}

@ARTICLE{dyergreenhill,
	AUTHOR = {M. Dyer and C. Greenhill},
	TITLE = {On {M}arkov chains for independent sets},
	JOURNAL = {J. of Algorithms},
	YEAR = {2000},
	VOLUME = {35},
	PAGES = {17--49}
}

@ARTICLE{eliahou,
	AUTHOR = {S. Eliahou},
	TITLE = {An algebraic criterion for a graph to be four-colourable},
	JOURNAL = {Aportaciones Matem\'aticas},
	YEAR = {1992},
	VOLUME = {6},
	PAGES = {3--27}
}

@INPROCEEDINGS{erdos,
	TITLE = {Results on the {E}rd\"os-{G}y\'arf\'as conjecture in $K_{1,m}$-free graphs},
	AUTHOR = {S.E. Shauger},
	BOOKTITLE = {Congressus Numerantium},
	ORGANIZATION = {Twenty-ninth Southeastern International
Conference on Combinatorics Graph Theory and Computing},
	YEAR = {1998},
	PAGES = {61--65}
}

@ARTICLE{fischer,
	AUTHOR = {K.G. Fischer},
	TITLE = {Symmetric polynomials and {H}all's theorem},
	JOURNAL = {Discrete Math},
	YEAR = {1988},
	VOLUME = {69},
	NUMBER = {3},
	PAGES = {225--234}
}

@BOOK{garey_and_johnson,
	AUTHOR = "M. Garey and D. Johnson",
	TITLE = "Computers and Intractability: A Guide to the Theory of NP-Completeness",
	PUBLISHER = {W.H. Freeman and Company},
	YEAR = "1979"
}

@BOOK{book_dom,
	AUTHOR = "T. W. Haynes and S. T. Hedetniemi and P. J. Slater",
	TITLE = "Domination in Graphs: Advanced Topics",
	PUBLISHER = {Chapman \& Hall},
	YEAR = "1998"
}

@BOOK{book_dom2,
	AUTHOR = "T. W. Haynes and S. T. Hedetniemi and P. J. Slater",
	TITLE = "Fundamentals of Domination in Graphs",
	PUBLISHER = {Chapman \& Hall},
	YEAR = "1998"
}


@ARTICLE{ho_total_dom,
	AUTHOR = {P. T. Ho},
	TITLE = {A note on the total domination number},
	JOURNAL = {Utilitas Mathematica},
	YEAR = {2008},
	VOLUME = {77},
	PAGES = {97--100}
}

@ARTICLE{hou_jiang_pr_dom,
	AUTHOR = {X. M. Hou and F. Jiang},
	TITLE = {Paired Domination of {C}artesian Products of Graphs},
	JOURNAL = {Journal of Mathematical Research \& Exposition},
	YEAR = {2010},
	VOLUME = {30},
	NUMBER = {1},
	PAGES = {181--185}
}

@ARTICLE{hou_lu,
	AUTHOR = {X. M. Hou and Y. Lu},
	TITLE = {On the \{k\}-domination number of {C}artesian products of graphs},
	JOURNAL = {Discrete Mathematics},
	YEAR = {2009},
	VOLUME = {309},
	PAGES = {3413--3419}
}


@ARTICLE{lili,
	AUTHOR = {S.R. Li and W.W. Li},
	TITLE = {Independence number of graphs and generators of ideals},
	JOURNAL = {Combinatorica},
	YEAR = {1981},
	VOLUME = {1},
	PAGES = {55--61}
}

@ARTICLE{lovasz1,
	AUTHOR = {L. Lov\'asz},
	TITLE = {Stable sets and polynomials},
	JOURNAL = {Discrete Mathematics},
	YEAR = {1994},
	VOLUME = {124},
	PAGES = {137--153}
}

@ARTICLE{mammen_hogg,
	AUTHOR = {D. Mammen and T. Hogg},
	TITLE = {A New Look at Easy-Hard-Easy Pattern of Cominatorial Search Difficulty},
	JOURNAL = {Journal of Artificial Intelligence Research},
	YEAR = {1997},
	VOLUME = {7},
	PAGES = {47--66}
}

@ARTICLE{monique07,
	AUTHOR = {M. Laurent},
	TITLE = {Semidefinite representations for
  finite varieties},
	JOURNAL = {Mathematical Programming},
	YEAR = {2007},
	VOLUME = {109},
	PAGES = {1--26}
}

@ARTICLE{mm,
	AUTHOR = {E.W. Mayr and A.R. Meyer},
	TITLE = {The Complexity of the Word Problems for Commutative Semigroups and Polynomial Ideals},
	JOURNAL = {Advances in Mathematics},
	YEAR = {1982},
	VOLUME = {45},
	PAGES = {305--329}
}

@ARTICLE{matiyasevich1,
	AUTHOR = {Y. Matiyasevich},
	TITLE = {A criteria for colorability of vertices stated in terms of edge orientations (in Russian)},
	JOURNAL = {Discrete Analysis (Novosibirsk)},
	YEAR = {1974},
	VOLUME = {26},
	PAGES = {65--71}
}

@ARTICLE{matiyasevich2,
	AUTHOR = {Y. Matiyasevich},
	TITLE = {Some algebraic methods for
  calculation of the number of colorings of a graph (in Russian)},
	JOURNAL = {Zapiski Nauchnykh Seminarov POMI},
	YEAR = {2001},
	VOLUME = {293},
	PAGES = {193--205}
}

@INPROCEEDINGS{mnuk,
	TITLE = {Representing graph properties by polynomial ideals},
	AUTHOR = {M. Mnuk},
	BOOKTITLE = {Computer Algebra in Scientific Computing},
	ORGANIZATION = {4th Inter. Work. on Comp. Alg.
in Sci. Comp.},
	YEAR = {2001},
	PAGES = {431-444}
}


@ARTICLE{mendez_zabala,
	AUTHOR = {I. M\'endez-D\'iaz and P. Zabala},
	TITLE = {A branch-and-cut algorithm for graph coloring},
	JOURNAL = {Discrete Applied Mathematics},
	YEAR = {2006},
	VOLUME = {154},
	NUMBER = {5},
	PAGES = {826--847}
}


@ARTICLE{montgomery,
	AUTHOR = {P. Montgomery},
	TITLE = {A block {L}anczos algorithm for finding dependencies over $\text{GF}(2)$},
	JOURNAL = {Lecture Notes in Computer Science},
	YEAR = {1995},
	VOLUME = {921},
	PAGES = {106--120}
}


@BOOK{rand_alg,
	AUTHOR = "R. Motwani and P. Raghavan",
	TITLE = "Randomized Algorithms",
	PUBLISHER = {Cambridge University Press},
	YEAR = "1995"
}


@ARTICLE{nm_hard,
	AUTHOR = {K. Mizuno and S. Nishihara},
	TITLE = {Constructive generation of very hard 3-colorability instances},
	JOURNAL = {Discrete Applied Mathematics},
	YEAR = {2008},
	VOLUME = {156},
	NUMBER = {2},
	PAGES = {218--229}
}

@ARTICLE{onn,
	AUTHOR = {K. Mizuno and S. Nishihara},
	TITLE = {Nowhere-zero flow polynomials},
	JOURNAL = {Journal of Combinatorial Theory, Series A},
	YEAR = {2004},
	VOLUME = {108},
	NUMBER = {2},
	PAGES = {205--215}
}


@BOOK{dimitriou,
	AUTHOR = "C. Papadimitriou",
	TITLE = "Computational Complexity",
	PUBLISHER = {Addison Wesley Longman},
	YEAR = "1994"
}

@ARTICLE{parrilo1,
	AUTHOR = {P. Parrilo},
	TITLE = {Semidefinite programming relaxations
  for semialgebraic problems},
	JOURNAL = {Mathematical Programming, Series B},
	YEAR = {2003},
	VOLUME = {96},
	NUMBER = {2},
	PAGES = {293--320}
}

@ARTICLE{picouleau,
	AUTHOR = {C. Picouleau},
	TITLE = {Complexity of the {H}amiltonian cycle in regular graph problem},
	JOURNAL = {Theoretical computer science},
	YEAR = {1994},
	VOLUME = {131},
	NUMBER = {2},
	PAGES = {463--473}
}

@ARTICLE{rostalski1,
   author = {J. Lasserre and M. Laurent and P. Rostalski},
   title = {Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals},
	JOURNAL = {Foundations of Computational Mathematics},
	YEAR = {2007},
}

@BOOK{schrijver,
	AUTHOR = "A. Schrijver",
	TITLE = "Theory of Linear and Integer Programming",
	PUBLISHER = {John Wiley \& Sons},
	YEAR = "1986",
	SERIES = {Wiley-Interscience Series in Discrete Mathematics and Optimization},
	ADDRESS = {Chichester, West Sussex, England}
}


@ARTICLE{schnyder,
	AUTHOR = {W. Schnyder},
	TITLE = {Planar graphs and poset dimension},
	JOURNAL = {Order},
	YEAR = {1989},
	VOLUME = {5},
	NUMBER = {4},
	PAGES = {323--343}
}

@BOOK{sipser,
	AUTHOR = "M. Sipser",
	TITLE = "Introduction to the Theory of Computation",
	PUBLISHER = {PWS Publishing Company},
	YEAR = "2001"
}

@ARTICLE{sum_blitch,
	AUTHOR = {D. Sumner and P. Blitch},
	TITLE = {Domination Critical Graphs},
	JOURNAL = {Journal of Cominatorial Theory, Series B},
	YEAR = {1983},
	VOLUME = {34},
	PAGES = {65--76}
}



@INPROCEEDINGS{susan_issac,
   author = {J.A. De Loera and J. Lee and P.N. Malkin and S. Margulies},
   title = {Hilbert's {N}ullstellensatz and an algorithm for proving combinatorial infeasibility},
	BOOKTITLE = {Proceedings of the twenty-first international symposium on Symbolic and algebraic computation},
	ORGANIZATION = {ISSAC},
	YEAR = {2008},
	PAGES = {197-206}
}

@ARTICLE{susan_cpc,
   author = {J.A. De Loera and J. Lee and S. Margulies and S. Onn},
   title = {Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the {N}ullstellensatz},
	JOURNAL = {Combinatorics, Probability and Computing},
	YEAR = {2009},
	VOLUME = {18},
	NUMBER = {4},
	PAGES = {551--582}
}

@article{susan_thesis,
   author = {S. Margulies},
   title = {Computer {A}lgebra, {C}ombinatorics, and {C}omplexity: {H}ilbert's {N}ullstellensatz and {NP}-{C}omplete {P}roblems},
   JOURNAL = {University of California at Davis, Ph.D. Thesis},
   YEAR = {2008}
}

@Misc{smallk,
   author = {J. Culberson},
   title = {Overview of the Smallk Graph Coloring Program},
   howpublished ={\url{http://www.cs.ualberta.ca/\~joe/Coloring/Colorscr/smallk.html788}},
}

@ARTICLE{villa1,
	AUTHOR = {A. Simis and W. Vasconcelos and R. Villarreal},
	TITLE = {On the ideal theory of graphs},
	JOURNAL = {Journal of Algebra},
	YEAR = {1994},
	VOLUME = {167},
	NUMBER = {2},
	PAGES = {389--416}
}

@article {vizing,
	author = {V. Vizing},
	title = {Some unsolved problems in graph theory},
	journal = {Uspekhi Mat. Nauk},
	volume = "23",
	year = "1968",
	pages = "117--134"
}

@article {viz_bar_ger,
	author = {A. Barcalkin and L. German},
	title = {The external stability number of the {C}artesian product of graphs},
	journal = {Bul. Akad. Stiinte RSS Moldoven.},
	volume = "1",
	number = "94",
	year = "1979",
	pages = "5--8"
}

@article{univ_grob,
	author = {E. Babson and S. Onn and R. Thomas},
	title = {The {H}ilbert zonotope and a polynomial time algorithm for universal {G}{\"o}rbner bases},
	journal = {Advances in Applied Mathematics},
	volume = "30",
	number = "3",
	year = "2003",
	pages = "529--544"
}

@INPROCEEDINGS{viz_dom_bk,
  	author =      "B. Hartnell and D. Rall",
  	title =       "Domination in {C}artesian products: {V}izing's conjecture",
  	booktitle =   "Domination in graphs",
	organization = "Mono. Textbooks Pure and Appl. Math",
	pages = "163--189",
  	publisher =   "Dekker",
  	address = "New York",
  	year =        1998
}

@article {viz_clark_suen,
	author = {W. Clark and S. Suen},
	title = {An inequality related to {V}izing's conjecture},
	journal = {Electronic Journal of Combinatorics},
	volume = "7",
	number = "Note 4",
	year = "2000"
}

@article{viz_survey_2009,
    author = {B. Bre\u{s}ar and P. Dorbec and W.Goddard and B.Hartnell and M.Henning and S.Klav\u{z}ar and D.Rall},
    title = {Vizing's conjecture: A survey and recent results},
journal = {Journal of Graph Theory},
	volume = "69",
	number = "1",
	year = "2012",
	pages = "46--76"
}

@article{rall_pr,
    author = {B. Bre\u{s}ar and M. A. Henning and  D. F. Rall},
    title = {Paired-domination of {C}artesian products of graphs},
	journal = {Utilitas Mathematica},
	volume = "73",
	year = "2007",
	pages = "255--265"
}

@article {viz_sun,
	author = {L. Sun},
	title = {A result on {V}izing's conjecture},
	journal = {Discrete Mathematics},
	volume = "275",
	number = "1",
	year = "2004",
	pages = "363--366"
}

@article{zahar_pareek,
	author = {M. El-Zahar and C.M. Pareek},
	title = {Domination number of products of graphs},
	journal = {Ars Combin.},
	volume = "31",
	year = "1991",
	pages = "223--227"
}

@INPROCEEDINGS{vlasie,
	TITLE = {Systematic generation of very hard cases for graph 3-colorability},
	AUTHOR = {R. Vlasie},
	BOOKTITLE = {Tools with Artificial Intelligence},
	ORGANIZATION = {Seventh International Conference on Artificial Intelligence},
	YEAR = {1995},
	PAGES = {114-119}
}

@article{yannakakis,
  author    = {M. Yannakakis},
  title     = {Expressing Combinatorial Optimization Problems by Linear
               Programs},
  journal   = {Journal of Computer and System Sciences},
  volume    = {43},
  number    = {3},
  year      = {1991},
  pages     = {441-466}
}
