  %number={02},
@conference{Brooks41,
  title={{On colouring the nodes of a network}},
  author={Brooks, R.L.},
  booktitle={Mathematical Proceedings of the Cambridge Philosophical Society},
  volume={37},
  pages={194--197},
  issn={0305-0041},
  year={1941},
  organization={Cambridge Univ Press}
}

@article{grötschel1981ellipsoid,
  title={{The ellipsoid method and its consequences in combinatorial optimization}},
  author={Gr\"{o}tschel, M. and Lov\'{a}sz, L. and Schrijver, A.},
  journal={Combinatorica},
  volume={1},
  number={2},
  pages={169--197},
  issn={0209-9683},
  year={1981},
  publisher={Springer}
}

@article{kierstead2009ore,
  title={{Ore-type versions of Brooks' theorem}},
  author={Kierstead, H.A. and Kostochka, A.V.},
  journal={Journal of Combinatorial Theory, Series B},
  volume={99},
  number={2},
  pages={298--305},
  issn={0095-8956},
  year={2009},
  publisher={Elsevier}
}

@article{mozhan1983,
  title={{Chromatic number of graphs with a density that does not exceed two-thirds of the maximal degree}},
  author={Mozhan, N.N.},
  journal={Metody Diskretn. Anal.},
  volume={39},
  pages={52--65 (in Russian)},
  year={1983}
}

@article{rabern2010a,
  title={{$\Delta$-critical graphs with small high vertex cliques}},
  author={Rabern, L.},
  journal={Journal of Combinatorial Theory, Series B},
  volume={102},
  number={1},
  pages={126--130},
  year={2012}
}

@article{rabern2010b,
  title={{An improvement on Brooks' theorem}},
  author={Rabern, L.},
  journal={Manuscript},
  year={2010}
}

@article{stacho2001new,
  title={{New upper bounds for the chromatic number of a graph}},
  author={Stacho, L.},
  journal={Journal of Graph Theory},
  volume={36},
  number={2},
  pages={117--120},
  issn={1097-0118},
  year={2001},
  publisher={John Wiley \& Sons}
}

@article{stiebitz1982proof,
  title={{Proof of a conjecture of T. Gallai concerning connectivity properties of colour-critical graphs}},
  author={Stiebitz, M.},
  journal={Combinatorica},
  volume={2},
  number={3},
  pages={315--323},
  issn={0209-9683},
  year={1982},
  publisher={Springer}
}

@article{krs_one,
  title={{Graphs with chromatic number close to maximum degree}},
  author={Kostochka, A.V. and Rabern, L. and Stiebitz, M.},
  journal={Discrete Mathematics},
  volume={312},
  number={6},
  pages={1273--1281},
  year={2012},
  publisher={Elsevier}
}

@article{reed1999strengthening,
  title={{A strengthening of Brooks' theorem}},
  author={Reed, B.},
  journal={Journal of Combinatorial Theory, Series B},
  volume={76},
  number={2},
  pages={136--149},
  issn={0095-8956},
  year={1999},
  publisher={Elsevier}
}

@article{king2008bounding,
  title={{Bounding $\chi$ in terms of $\omega$ and $\Delta$ for quasi-line graphs}},
  author={King, A.D. and Reed, B.A.},
  journal={Journal of Graph Theory},
  volume={59},
  number={3},
  pages={215--228},
  issn={1097-0118},
  year={2008},
  publisher={Wiley Online Library}
}

@phdthesis{king2009claw,
  title={{Claw-free graphs and two conjectures on omega, Delta, and chi}},
  author={King, A.},
  year={2009},
  school={McGill University}
}

@article{rabernhitting,
  title={{On hitting all maximum cliques with an independent set}},
  author={Rabern, L.},
  journal={Journal of Graph Theory},
  volume={66},
  number={1},
  pages={32--37},
  year={2011},
  publisher={Wiley Online Library}
}

@article{king2007upper,
  title={{An upper bound for the chromatic number of line graphs}},
  author={King, A.D. and Reed, B.A. and Vetta, A.},
  journal={European Journal of Combinatorics},
  volume={28},
  number={8},
  pages={2182--2187},
  issn={0195-6698},
  year={2007},
  publisher={Elsevier}
}

@book{molloy2002graph,
  title={{Graph colouring and the probabilistic method}},
  author={Molloy, M.S. and Reed, B.A.},
  isbn={3540421394},
  year={2002},
  publisher={Springer Verlag}
}

@article{KingHitting,
  title={Hitting all maximum cliques with a stable set using lopsided independent transversals},
  author={King, A.D.},
  journal={Journal of Graph Theory},
  year={2010},
  publisher={Wiley Online Library}
}

    %journal={Arxiv preprint, arXiv:0907.3705},
@ARTICLE{2009arXiv0907.3705R,
   author = {Rabern, L.},
    title = {{On hitting all maximum cliques with an independent set}},
    journal={Arxiv preprint, \url{http://arxiv.org/abs/0907.3705}},
    year={2009}
}

@article{BorodinK77,
  title={{On an upper bound of a graph's chromatic number, depending on the graph's degree and density}},
  author={Borodin, O.V. and Kostochka, A.V.},
  fjournal={Journal of Combinatorial Theory, Series B},
  journal={J. of Comb. Theory, B},
  volume={23},
  number={2-3},
  pages={247--250},
  issn={0095-8956},
  year={1977},
  publisher={Elsevier}
}

@article{kostochkaRussian,
  title={{Degree, density, and chromatic number}},
  author={Kostochka, A.V.},
  journal={Metody Diskret. Anal.},
  volume={35},
  pages={45--70 (in Russian)},
  year={1980},
}

@article{reed1998omega,
  title={{$\omega$, $\Delta$, and $\chi$}},
  author={Reed, B.},
  journal={Journal of Graph Theory},
  volume={27},
  number={4},
  pages={177--212},
  issn={1097-0118},
  year={1998},
  publisher={Wiley Online Library}
}}

@article{ReedSudakov,
  title={{List colouring when the chromatic number is close to the order of the graph}},
  author={Reed, B. and Sudakov, B.},
  journal={Combinatorica},
  volume={25},
  number={1},
  pages={117--123},
  issn={0209-9683},
  year={2004},
  publisher={Springer}
}

@article{BrooksExtended,
  title={{The colour theorems of Brooks and Gallai extended}},
  author={Kostochka, A.V. and Stiebitz, M. and Wirth, B.},
  journal={Discrete Mathematics},
  volume={162},
  number={1-3},
  pages={299--303},
  year={1996},
  publisher={Elsevier}
}

@article{HladkyKS10,
  title={{Brooks' Theorem via the Alon-Tarsi Theorem}},
  author={Hladk{\`y}, J. and Kr{\'a}l, D. and Schauz, U.},
  journal={Discrete Mathematics},
  year={2010},
  publisher={Elsevier}
}

@conference{ErdosRT79,
  title={{Choosability in graphs}},
  author={Erd\H{o}s, P. and Rubin, A.L. and Taylor, H.},
  booktitle={Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium},
  volume={26},
  pages={125--157},
  year={1979}
}

@conference{borodin1977criterion,
  title={{Criterion of chromaticity of a degree prescription}},
  author={Borodin, O.V.},
  booktitle={Abstracts of IV All-Union Conf. on Th. Cybernetics},
  location={Novosibirsk},
  pages={127--128},
  year={1977}
}

@ARTICLE{Alon19997,
author={Alon, N.},
title={Combinatorial Nullstellensatz},
journal={Combinatorics Probability and Computing},
year={1999},
volume={8},
number={1--2},
pages={7-29}
}

% Where Alon-Tarsi Theorem was proved
@ARTICLE{AlonT92,
author={Alon, N. and Tarsi, M.},
title={Colorings and orientations of graphs},
journal={Combinatorica},
year={1992},
volume={12},
number={2},
pages={125--134}
}

@incollection{Entringer1985367,
title = {{A Short Proof of Rubin's Block Theorem}},
editor = {B.R. Alspach and C.D. Godsil},
booktitle = {{Annals of Discrete Mathematics 27 - Cycles in Graphs}},
publisher = {North-Holland},
year = {1985},
volume = {115},
pages = {367--368},
series = {North-Holland Mathematics Studies},
author = {Entringer, R.C.}
}

@ARTICLE{Lovasz1975269,
author={Lov\'{a}sz, L.},
title={Three short proofs in graph theory},
journal={Journal of Combinatorial Theory, Series B},
year={1975},
volume={19},
number={3},
pages={269--271},
}

@article{kierstead2000choosability,
  title={{On the choosability of complete multipartite graphs with part size three}},
  author={Kierstead, H.A.},
  journal={Discrete Mathematics},
  volume={211},
  number={1-3},
  pages={255--259},
  issn={0012-365X},
  year={2000},
  publisher={Elsevier}
}

@article{dhurandhar1982improvement,
  title={Improvement on {Brooks}' chromatic bound for a class of graphs},
  author={Dhurandhar, M.},
  journal={Discrete Mathematics},
  volume={42},
  number={1},
  pages={51--56},
  year={1982},
  publisher={Elsevier}
}

@article{kierstead1986chromatic,
  title={The chromatic number of graphs which induce neither {$K_{1,3}$} nor {$K_5-e$}},
  author={Kierstead, H.A. and Schmerl, J.H.},
  journal={Discrete mathematics},
  volume={58},
  number={3},
  pages={253--262},
  year={1986},
  publisher={Elsevier}
}

@article{GoldbergJGT,
  title={Edge-coloring of multigraphs: Recoloring technique},
  author={Goldberg, M.K.},
  journal={Journal of Graph Theory},
  volume={8},
  number={1},
  pages={123--137},
  year={1984},
  publisher={Wiley Online Library}
}

@article{StiebitzVizingGoldberg,
  title={Graph edge colouring: Vizing's theorem and Goldberg's conjecture},
  author={Favrholdt, L.M. and Stiebitz, M. and Toft, B.},
  year={2006},
  publisher={University of Southern Denmark, preprint}
}

@article{catlin1979hajos,
  title={Haj{\'o}s’ graph-coloring conjecture: variations and counterexamples},
  author={Catlin, P.A.},
  journal={J. Combin. Theory Ser. B},
  volume={26},
  number={2},
  pages={268--274},
  year={1979}
}

@book{GraphColoringProblems,
  title={Graph coloring problems},
  author={Jensen, T.R. and Toft, B.},
  year={1995},
  publisher={John Wiley \& Sons}
}

@article{anderson1977edge,
  title={On edge-colorings of graphs},
  author={Andersen, L.D.},
  journal={Math. Scand},
  volume={40},
  pages={161--175},
  year={1977}
}

@article{cariolaro2006fans,
  title={On fans in multigraphs},
  author={Cariolaro, D.},
  journal={Journal of Graph Theory},
  volume={51},
  number={4},
  pages={301--318},
  year={2006},
  publisher={Wiley Online Library}
}

@article{gallai1963kritische,
  title={Kritische graphen I.},
  author={Gallai, T.},
  journal={Math. Inst. Hungar. Acad. Sci},
  volume={8},
  pages={165--192 (in German)},
  year={1963}
}

@article{chudnovsky2005structure,
  title={The structure of claw-free graphs},
  author={Chudnovsky, M. and Seymour, P.},
  journal={Surveys in combinatorics},
  volume={327},
  pages={153--171},
  year={2005}
}

@article{rabern2011strengthening,
  title={{A strengthening of Brooks' Theorem for line graphs}},
  author={Rabern, L.},
  journal={Electron. J. Combin.},
  volume={18},
  number={p145},
  pages={1},
  year={2011}
}

@article{rabern2010destroying,
  title={Destroying non-complete regular components in graph partitions},
  author={Rabern, L.},
  journal={J. Graph Theory},
  year={In Press}
}

@article{partitionnote,
  title={A note on vertex partitions},
  author={Rabern, L.},
  journal={Manuscript},
  year={2011}
}

@article{lovasz1966decomposition,
  title={On decomposition of graphs},
  author={Lovasz, L.},
  journal={SIAM J Algebraic and Discrete Methods},
  volume={3},
  number={1},
  pages={237--238},
  year={1966},
  publisher={SIAM}
}

@article{CatlinAnotherBound,
  title={{Another bound on the chromatic number of a graph}},
  author={Catlin, P.A.},
  journal={Discrete Mathematics},
  volume={24},
  number={1},
  pages={1--6},
  year={1978},
  publisher={Elsevier}
}

@article{KostochkaTriangleFree,
  title={{A modification of Catlin's algorithm}},
  author={Kostochka, A.V.},
  journal={Methods and Programs of Solutions Optimization Problems on Graphs and Networks},
  number={2},
  pages={75--79 (in Russian)},
  year={1982}
}

@article{bollobasManvel,
  title={Optimal vertex partitions},
  author={Bollob{\'a}s, B. and Manvel, B.},
  journal={Bulletin of the London Mathematical Society},
  volume={11},
  number={2},
  pages={113},
  year={1979},
  publisher={Oxford University Press}
}

@article{catlin1978bound,
  title={A bound on the chromatic number of a graph},
  author={Catlin, P.A.},
  journal={Discrete Mathematics},
  volume={22},
  number={1},
  pages={81--83},
  year={1978},
  publisher={Elsevier}
}

@article{lawrence1978covering,
  title={Covering the vertex set of a graph with subgraphs of smaller degree},
  author={Lawrence, J.},
  journal={Discrete Mathematics},
  volume={21},
  number={1},
  pages={61--68},
  year={1978},
  publisher={Elsevier}
}

@article{haxell2001note,
  title={A note on vertex list colouring},
  author={Haxell, PE},
  journal={Combinatorics, Probability and Computing},
  volume={10},
  number={04},
  pages={345--347},
  year={2001},
  publisher={Cambridge Univ Press}
}

@article{aharoni2007independent,
  title={Independent systems of representatives in weighted graphs},
  author={Aharoni, R. and Berger, E. and Ziv, R.},
  journal={Combinatorica},
  volume={27},
  number={3},
  pages={253--267},
  year={2007},
  publisher={Springer}
}

@article{HajnalSaturation,
  title={{A theorem on $k$-saturated graphs}},
  author={Hajnal, A.},
  journal={Canadian Journal of Mathematics},
  volume={17},
  number={5},
  pages={720},
  year={1965},
  publisher={Canadian Mathematical Society}
}

@article{niessen2000round,
  title={The round-up property of the fractional chromatic number for proper circular arc graphs},
  author={Niessen, T. and Kind, J.},
  journal={Journal of Graph Theory},
  volume={33},
  number={4},
  pages={256--267},
  year={2000},
  publisher={Wiley Online Library}
}

@article{chudnovskyFradkin,
  title={Coloring quasi-line graphs},
  author={Chudnovsky, M. and Ovetsky, A.},
  journal={Journal of Graph Theory},
  volume={54},
  number={1},
  pages={41--50},
  year={2007},
  publisher={Wiley Online Library}
}

@article{rabern2012partitioning,
  title={Partitioning and coloring with degree constraints},
  author={Rabern, L.},
  journal={Arxiv preprint, \url{http://arxiv.org/abs/1202.5855}},
  year={2012}
}

@article{tverberg1983brooks,
  title={{On Brooks' theorem and some related results}},
  author={Tverberg, H.},
  journal={Mathematics Scandinavia},
  volume={52},
  pages={37--40},
  year={1983}
}

@article{haxell2006odd,
  title={Odd independent transversals are odd},
  author={Haxell, P. and Szab{\'o}, T.},
  journal={Combinatorics Probability and Computing},
  volume={15},
  number={1/2},
  pages={193},
  year={2006},
  publisher={Cambridge University Press; 1999}
}

@article{haxell2011forming,
  title={On forming committees},
  author={Haxell, P.},
  journal={The American Mathematical Monthly},
  volume={118},
  number={9},
  pages={777--788},
  year={2011},
  publisher={JSTOR}
}

@article{haxell2004strong,
  title={On the strong chromatic number},
  author={Haxell, P.},
  journal={Combinatorics, Probability and Computing},
  volume={13},
  number={06},
  pages={857--865},
  year={2004},
  publisher={Cambridge Univ Press}
}


    %AUTHOR = {Cranston, Daniel W. and Rabern, Landon},
@article{CranstonR13claw,
    author={Cranston, Daniel W. and Rabern, Landon},
     TITLE = {Coloring claw-free graphs with {$\Delta-1$} colors},
   JOURNAL = {SIAM J. Discrete Math.},
  FJOURNAL = {SIAM Journal on Discrete Mathematics},
    VOLUME = {27},
      YEAR = {2013},
    NUMBER = {1},
     PAGES = {534--549},
      ISSN = {0895-4801},
     CODEN = {SJDMEC},
   MRCLASS = {05C15},
  MRNUMBER = {3032933},
       DOI = {10.1137/12088015X},
       URL = {http://dx.doi.org/10.1137/12088015X},
}

@article{CranstonR13+big,
    AUTHOR = {Cranston, Daniel W. and Rabern, Landon},
     TITLE = {Graphs with {$\chi=\Delta$} have big cliques},
   JOURNAL = {SIAM J. Discrete Math.},
  FJOURNAL = {SIAM Journal on Discrete Mathematics},
    VOLUME = {29},
      YEAR = {2015},
    NUMBER = {4},
     PAGES = {1792--1814},
      ISSN = {0895-4801},
   MRCLASS = {05C85 (05C15)},
  MRNUMBER = {3403134},
       DOI = {10.1137/130929515},
  note ={Preprint available at \arxiv{1305.3526}},
}

@article{denseneighborhoods,
  title={{Coloring graphs with dense neighborhoods}},
  author={Rabern, L.},
  journal={ J. Graph Theory, Forthcoming},
}

@article{beutelspacher1984minimal,
  title={Minimal graphs for which the chromatic number equals the maximal degree},
  author={Beutelspacher, A. and Hering, P.R.},
  journal={Ars Combin},
  volume={18},
  pages={201--216},
  year={1984}
}

@article{BorodinKostochkaWoodall,
  author = {Borodin, O. V. and Kostochka, A. V. and Woodall, D. R.},
  title = {List edge and list total colourings of multigraphs},
  journal = {J. Combin. Theory Ser. B},
  volume = {71},
  number = {2},
  pages = {184--204},
  year = {1997},
}

@Misc{PersonalComms,
  author = {Kostochka, A. V.},
  howpublished = {personal communication},
  year = 2012
}

@Misc{KingPersonalKostochkaGeneralized,
  author = {King, A.},
  howpublished = {personal communication},
  year = 2009
}

@article{gravier1998graphs,
  title={Graphs whose choice number is equal to their chromatic number},
  author={Gravier, S. and Maffray, F.},
  journal={Journal of Graph Theory},
  volume={27},
  number={2},
  pages={87--97},
  year={1998},
  publisher={Wiley Online Library}
}

@article{Schauz09,
  title={{Mr. Paint and Mrs. Correct}},
  author={Schauz, U.},
  journal={Electron. J. Combin.},
  fjournal={{The Electronic Journal of Combinatorics}},
  volume={16},
  number={1},
  pages={R77},
  year={2009}
}

@article{szabo2006extremal,
  title={Extremal problems for transversals in graphs with bounded degree},
  author={Szab{\'o}, T. and Tardos, G.},
  journal={Combinatorica},
  volume={26},
  number={3},
  pages={333--351},
  year={2006},
  publisher={Springer}
}

@article{mules,
  AUTHOR = {Cranston, Daniel W. and Rabern, Landon},
     TITLE = {Coloring a graph with {$\Delta-1$} colors: conjectures
              equivalent to the {B}orodin-{K}ostochka conjecture that appear
              weaker},
   JOURNAL = {European J. Combin.},
  FJOURNAL = {European Journal of Combinatorics},
    VOLUME = {44},
      YEAR = {2015},
    NUMBER = {part A},
     PAGES = {23--42},
      ISSN = {0195-6698},
   MRCLASS = {05C15},
MRREVIEWER = {M. Venkatachalam},
      note ={Preprint available at \arxiv{1203.5380}},
}

@inproceedings{mader,
  title={{Existenz n-fach zusammenh{\"a}ngender Teilgraphen in Graphen gen{\"u}gend gro{\ss}er Kantendichte}},
  author={Mader, W.},
  booktitle={{Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg}},
  volume={37},
  number={1},
  pages={86--97},
  year={1972},
  organization={Springer}
}

@BOOK{diestel2010,
	AUTHOR       = {Diestel, R.},
	TITLE        = {{Graph Theory}},
	PUBLISHER    = {Springer-Verlag, Heidelberg},
	YEAR         = {2010},
	EDITION      = {4}
}

@article{kostochkastiebitzedgesincriticalgraph,
  title={A new lower bound on the number of edges in colour-critical graphs and hypergraphs},
  author={Kostochka, A.V. and Stiebitz, M.},
  journal={Journal of Combinatorial Theory, Series B},
  volume={87},
  number={2},
  pages={374--402},
  year={2003},
  publisher={Elsevier}
}

@article{Schauz-AT,
  title={{Flexible color lists in Alon and Tarsi's theorem, and time scheduling with unreliable participants}},
  author={Schauz, U.},
  journal={Electron. J. Combin.},
  fjournal={The Electronic Journal of Combinatorics},
  volume={17},
  number={1},
  pages={R13},
  year={2010}
}

@article{dirac1951note,
  title={Note on the colouring of graphs},
  author={Dirac, G.A.},
  journal={Mathematische Zeitschrift},
  volume={54},
  number={4},
  pages={347--353},
  year={1951},
  publisher={Springer}
}

@article{dirac1957theorem,
  title={{A theorem of R.L. Brooks and a conjecture of H. Hadwiger}},
  author={Dirac, G.A.},
  journal={Proceedings of the London Mathematical Society},
  volume={3},
  number={1},
  pages={161--195},
  year={1957},
  publisher={Oxford University Press}
}

@article{dirac1974number,
  title={The number of edges in critical graphs},
  author={Dirac, G.A.},
  journal={J. Reine Angew. Math},
  volume={268},
  number={269},
  pages={150--164},
  year={1974}
}

@article{krivelevich1997minimal,
  title={On the minimal number of edges in color-critical graphs},
  author={Krivelevich, M.},
  journal={Combinatorica},
  volume={17},
  number={3},
  pages={401--426},
  year={1997},
  publisher={Springer}
}

@article{kostochkayancey2012ore,
  title={{Ore's Conjecture on color-critical graphs is almost true}},
  author={Kostochka, A. and Yancey, M.},
  journal={Arxiv preprint, \url{http://arxiv.org/abs/1209.1050}},
  year={2012}
}

@book{ore1967four,
  title={The four-color problem},
  author={Ore, O.},
  volume={27},
  year={1967},
  publisher={Academic Press}
}

@article{kostochka2012oregrotsch,
  title={{Ore's Conjecture for $k=4$ and Gr\"otzsch Theorem}},
  author={Kostochka, A. and Yancey, M.},
  journal={Arxiv preprint, \url{http://arxiv.org/abs/1209.1173}},
  year={2012}
}

@article{Zhu09online,
  title={On-line list colouring of graphs},
  author={Zhu, X.},
  journal={Electron. J. Combin.},
  fjournal={{The Electronic Journal of Combinatorics}},
  volume={16},
  number={1},
  pages={R127},
  year={2009}
}

@article{Vizing76,
  title={{Vextex coloring with given colors}},
  author={Vizing, V.G.},
  journal={Metody Diskretn. Anal.},
  volume={29},
  pages={3--10 (in Russian)},
  year={1976}
}

@article{riasat2012critically,
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