Graph and hypergraph colouring via nibble methods: A survey

This paper provides a survey of methods, results, and open problems on graph and hypergraph colourings, with a particular emphasis on semi-random ‘nibble’ methods. We also give a detailed sketch of some aspects of the recent proof of the Erdős-Faber-Lovász conjecture.

[1]  P. Alam.  ‘A’ , 2021, Composites Engineering: An A–Z Guide.

[2]  Mariusz Meszka.  The Chromatic Index of Projective Triple Systems , 2013 .

[3]  Jeff Kahn,et al.  Coloring Nearly-Disjoint Hypergraphs with n+o(n) Colors , 1992, J. Comb. Theory, Ser. A.

[4]  Ebadollah S. Mahmoodian,et al.  On the chromatic number of Latin square graphs , 2015, Discret. Math..

[5]  Dhruv Mubayi,et al.  Coloring Sparse Hypergraphs , 2014, SIAM J. Discret. Math..

[6]  Jeff Kahn,et al.  Fractional v. Integral Covers in Hypergraphs of Bounded Edge Size , 1997, J. Comb. Theory, Ser. A.

[7]  PETER NELSON,et al.  BOUNDING χ BY A FRACTION OF ∆ FOR GRAPHS WITHOUT LARGE CLIQUES , 2018 .

[8]  Vance Faber.  The Erdös-Faber-Lovász conjecture – the uniform regular case , 2010 .

[9]  R. L. Brooks.  On colouring the nodes of a network , 1941, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  D. K. Ray-Chaudhuri,et al.  Solution of Kirkman''s schoolgirl problem , 1971 .

[11]  A. D. Forbes,et al.  On independent sets , 2005 .

[12]  Alan M. Frieze,et al.  Coloring simple hypergraphs , 2013, J. Comb. Theory, Ser. B.

[13]  Tom Bohman,et al.  A natural barrier in random greedy hypergraph matching , 2019, Comb. Probab. Comput..

[14]  Daniela Kühn,et al.  Decompositions into isomorphic rainbow spanning trees , 2019, J. Comb. Theory, Ser. B.

[15]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[16]  János Komlós,et al.  Blow-up Lemma , 1997, Combinatorics, Probability and Computing.

[17]  Zoltán Füredi,et al.  The chromatic index of simple hypergraphs , 1986, Graphs Comb..

[18]  Luke Postle,et al.  Colouring Graphs with Sparse Neighbourhoods: Bounds and Applications , 2018, J. Comb. Theory B.

[19]  Charles J. Colbourn,et al.  Steiner Triple Systems with High Chromatic Index , 2017, SIAM J. Discret. Math..

[20]  Paul D. Seymour,et al.  A fractional version of the Erdős-Faber-Lovász conjecture , 1992, Comb..

[21]  Jeff Kahn,et al.  Asymptotics of the list-chromatic index for multigraphs , 2000, Random Struct. Algorithms.

[22]  Noga Alon,et al.  Asymmetric List Sizes in Bipartite Graphs , 2021, Annals of Combinatorics.

[23]  Gary Chartrand,et al.  Erdős on Graphs : His Legacy of Unsolved Problems , 2011 .

[24]  Will Perkins,et al.  Independent sets, matchings, and occupancy fractions , 2015, J. Lond. Math. Soc..

[25]  Vojtech Rödl,et al.  Near Perfect Coverings in Graphs and Hypergraphs , 1985, Eur. J. Comb..

[26]  Vojtech Rödl,et al.  On a Packing and Covering Problem , 1985, Eur. J. Comb..

[27]  Jeff Kahn,et al.  Asymptotics of the Chromatic Index for Multigraphs , 1996, J. Comb. Theory, Ser. B.

[28]  Yi Zhao,et al.  Recent advances on Dirac-type problems for hypergraphs , 2015, 1508.06170.

[29]  C. Shannon.  A Theorem on Coloring the Lines of a Network , 1949 .

[30]  Jeff Kahn,et al.  On Some Hypergraph Problems of Paul Erdős and the Asymptotics of Matchings, Covers and Colorings , 2013, The Mathematics of Paul Erdős I.

[31]  Jeff Kahn.  A linear programming perspective on the Frankl - Rödl - Pippenger theorem , 1996, Random Struct. Algorithms.

[32]  Benny Sudakov,et al.  Number of 1-Factorizations of Regular High-Degree Graphs , 2020, Comb..

[33]  Vojtech Rödl,et al.  Asymptotic packing and the random greedy algorithm , 1996, Random Struct. Algorithms.

[34]  B. Reed.  Graph Colouring and the Probabilistic Method , 2001 .

[35]  Michael Molloy,et al.  Asymptotically good edge correspondence colouring , 2018, 1808.08594.

[36]  Bruce A. Reed.  The list colouring constants , 1999, J. Graph Theory.

[37]  V. Sós,et al.  Irregularities of partitions , 1989 .

[38]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[39]  Gil Kalai,et al.  Some old and new problems in combinatorial geometry I: around Borsuk's problem , 2015, Surveys in Combinatorics.

[40]  Vojtech Rödl,et al.  Dirac-Type Questions For Hypergraphs — A Survey (Or More Problems For Endre To Solve) , 2010 .

[41]  C. Berge.  On the Chromatic Index of a Linear Hypergraph and the Chvátal Conjecture , 1989 .

[42]  Eugene L. Lawler,et al.  Edge coloring of hypergraphs and a conjecture of ErdÖs, Faber, Lovász , 1988, Comb..

[43]  V. Dara,et al.  A Proof of the Erdös - Faber - Lovász Conjecture , 2015 .

[44]  Alan M. Frieze,et al.  On the independence and chromatic numbers of random regular graphs , 1992, J. Comb. Theory, Ser. B.

[45]  Robert Morris,et al.  The Triangle-Free Process and the Ramsey Number 𝑅(3,𝑘) , 2020 .

[46]  Ross J. Kang,et al.  An improved procedure for colouring graphs of bounded local density , 2021, SODA.

[47]  S. Stein.  TRANSVERSALS OF LATIN SQUARES AND THEIR GENERALIZATIONS , 1975 .

[48]  R. Graham,et al.  On embedding graphs in squashed cubes , 1972 .

[49]  Daniela Kühn,et al.  A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree , 2021, ArXiv.

[50]  B. Roberts,et al.  On the average size of independent sets in triangle-free graphs , 2016, 1606.01043.

[51]  David A. Grable,et al.  More-Than-Nearly-Perfect Packings and Partial Designs , 1999, Comb..

[52]  János Komlós,et al.  A Dense Infinite Sidon Sequence , 1981, Eur. J. Comb..

[53]  Imre Z. Ruzsa,et al.  An Infinite Sidon Sequence , 1998 .

[54]  David A. Grable.  Nearly-Perfect Hypergraph Packing is in NC , 1996, Inf. Process. Lett..

[55]  N. Wormald.  The differential equation method for random graph processes and greedy algorithms , 1999 .

[56]  Rémi de Joannis de Verclos,et al.  Coloring triangle‐free graphs with local list sizes , 2020, Random Struct. Algorithms.

[57]  L. Postle,et al.  Edge-colouring graphs with local list sizes , 2020, 2007.14944.

[58]  Noga Alon,et al.  Coloring Graphs with Sparse Neighborhoods , 1999, J. Comb. Theory B.

[59]  D. Saxton,et al.  Hypergraph containers , 2012, 1204.6595.

[60]  Leslie Hogben,et al.  Combinatorial Matrix Theory , 2013 .

[61]  Wenan Zang,et al.  Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs , 2019, 1901.10316.

[62]  Noga Alon.  Independence numbers of locally sparse graphs and a Ramsey type problem , 1996, Random Struct. Algorithms.

[63]  Peter Keevash.  HYPERGRAPH MATCHINGS AND DESIGNS , 2018, Proceedings of the International Congress of Mathematicians (ICM 2018).

[64]  Javier Cilleruelo.  Infinite Sidon sequences , 2012 .

[65]  P. Hall.  On Representatives of Subsets , 1935 .

[66]  Claude Tardif,et al.  A clone-theoretic formulation of the Erdös-Faber-Lovász conjecture , 2004, Discuss. Math. Graph Theory.

[67]  Paul D. Seymour,et al.  Packing nearly-disjoint sets , 1982, Comb..

[68]  Mariusz Meszka,et al.  CIRCULANTS AND THE CHROMATIC INDEX OF STEINER TRIPLE SYSTEMS , 2004 .

[69]  Tom Bohman,et al.  Dynamic concentration of the triangle‐free process , 2013, Random Struct. Algorithms.

[70]  J. Kahn.  Asymptotics of Hypergraph Matching, Covering and Coloring Problems , 1995 .

[71]  David G. Harris,et al.  Edge-coloring linear hypergraphs with medium-sized edges , 2019, Random Struct. Algorithms.

[72]  Dong Yeap Kang,et al.  New bounds on the size of nearly perfect matchings in almost regular hypergraphs , 2020, Journal of the London Mathematical Society.

[73]  Zoltán Füredi,et al.  Matchings and covers in hypergraphs , 1988, Graphs Comb..

[74]  Ross J. Kang,et al.  An algorithmic framework for colouring locally sparse graphs , 2020, ArXiv.

[75]  D. Kuhn,et al.  Resolution of the Oberwolfach problem , 2018, Journal of the European Mathematical Society.

[76]  Jeong Han Kim,et al.  The Ramsey Number R(3, t) Has Order of Magnitude t2/log t , 1995, Random Struct. Algorithms.

[77]  Joel H. Spencer.  Asymptotic Packing via a Branching Process , 1995, Random Struct. Algorithms.

[78]  A. Brouwer.  On the Size of a Maximum Transversal in a Steiner Triple System , 1981, Canadian Journal of Mathematics.

[79]  Vojtech Rödl,et al.  On Uncrowded Hypergraphs , 1995, Random Struct. Algorithms.

[80]  V. G. Vizing.  The chromatic class of a multigraph , 1965 .

[81]  Van H. Vu,et al.  New bounds on nearly perfect matchings in hypergraphs: Higher codegrees do help , 2000, Random Struct. Algorithms.

[82]  Daniela Kühn,et al.  Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments , 2012, ArXiv.

[83]  de Ng Dick Bruijn.  A combinatorial problem , 1946 .

[84]  J. Kahn,et al.  A counterexample to Borsuk's conjecture , 1993, math/9307229.

[85]  T. Tao,et al.  Long gaps between primes , 2014, 1412.5029.

[86]  Gonzalo Fiz Pontiveros,et al.  The triangle-free process and R(3,k) , 2013 .

[87]  Peter Keevash.  The existence of designs , 2014, 1401.3665.

[88]  Vojtech Rödl,et al.  Partial Steiner systems and matchings in hypergraphs , 1998, Random Struct. Algorithms.

[89]  Anton Bernshteyn,et al.  The Johansson‐Molloy theorem for DP‐coloring , 2017, Random Struct. Algorithms.

[90]  Van H. Vu,et al.  A General Upper Bound on the List Chromatic Number of Locally Sparse Graphs , 2002, Combinatorics, Probability and Computing.

[91]  Zolt'an L'or'ant Nagy,et al.  Coloring linear hypergraphs: the Erdős–Faber–Lovász conjecture and the Combinatorial Nullstellensatz , 2020, Designs, Codes and Cryptography.

[92]  János Komlós,et al.  On Turán’s theorem for sparse graphs , 1981, Comb..

[93]  Noga Alon,et al.  Degrees and choice numbers , 2000, Random Struct. Algorithms.

[94]  W. T. Tutte.  The Factorization of Linear Graphs , 1947 .

[95]  Hao Huang,et al.  A counterexample to the Alon-Saks-Seymour conjecture and related problems , 2010, Comb..

[96]  Fotis Iliopoulos.  Improved Bounds for Coloring Locally Sparse Hypergraphs , 2021, APPROX-RANDOM.

[97]  D. de Werra,et al.  Graph Coloring Problems , 2013 .

[98]  Alan M. Frieze,et al.  On the Chromatic Number of Simple Triangle-Free Triple Systems , 2008, Electron. J. Comb..

[99]  Jeong Han Kim.  On Brooks' Theorem for Sparse Graphs , 1995, Comb. Probab. Comput..

[100]  Noga Alon,et al.  On the Degree, Size, and Chromatic Index of a Uniform Hypergraph , 1997, J. Comb. Theory, Ser. A.

[101]  Andrey Kupavskii,et al.  Rainbow structures in locally bounded colorings of graphs , 2018 .

[102]  P. Alam,et al.  R , 1823, The Herodotus Encyclopedia.

[103]  Uwe Schauz,et al.  Proof of the List Edge Coloring Conjecture for Complete Graphs of Prime Degree , 2014, Electron. J. Comb..

[104]  Noga Alon,et al.  On a Hypergraph Matching Problem , 2005, Graphs Comb..

[105]  E. Szemerédi,et al.  A Lower Bound for Heilbronn'S Problem , 1982 .

[106]  Dhruv Mubayi,et al.  Sparse hypergraphs with low independence number , 2013, Comb..

[107]  Joel H. Spencer,et al.  Asymptotic behavior of the chromatic index for hypergraphs , 1989, J. Comb. Theory, Ser. A.

[108]  Alistair Sinclair,et al.  Beyond the Lovász Local Lemma: Point to Set Correlations and Their Algorithmic Applications , 2018, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[109]  D. Kuhn,et al.  Surveys in Combinatorics 2009: Embedding large subgraphs into dense graphs , 2009, 0901.3541.

[110]  Deryk Osthus,et al.  Solution to a problem of Erd\H{o}s on the chromatic index of hypergraphs with bounded codegree , 2021, 2110.06181.

[112]  Gábor Tardos,et al.  A constructive proof of the general lovász local lemma , 2009, JACM.

[113]  P. Kayll,et al.  Two Chromatic Conjectures: One for Vertices and One for Edges , 2016 .

[114]  Zoltán Füredi,et al.  On the fractional matching polytope of a hypergraph , 1993, Comb..

[115]  Henning Bruhn,et al.  A stronger bound for the strong chromatic index , 2015, Electron. Notes Discret. Math..

[116]  Benny Sudakov,et al.  New bounds for Ryser’s conjecture and related problems , 2020, Transactions of the American Mathematical Society, Series B.

[117]  János Komlós,et al.  A Note on Ramsey Numbers , 1980, J. Comb. Theory, Ser. A.

[118]  Bruce A. Reed,et al.  Near-optimal list colorings , 2000, Random Struct. Algorithms.

[119]  Paul Erdös,et al.  On the combinatorial problems which I would most like to see solved , 1981, Comb..

[120]  Charles J. Colbourn,et al.  Hanani triple systems , 1993 .

[121]  Jeong Han Kim,et al.  Nearly perfect matchings in regular simple hypergraphs , 1997 .

[122]  Robin J. Wilson,et al.  Edge-colourings of graphs , 1977 .

[123]  Michael Molloy,et al.  The list chromatic number of graphs with small clique number , 2017, J. Comb. Theory B.

[124]  Daniela Kühn,et al.  The existence of designs via iterative absorption: hypergraph $F$-designs for arbitrary $F$. , 2020 .

[125]  Noga Alon,et al.  The choice number of random bipartite graphs , 1998 .

[126]  James B. Shearer,et al.  On the Independence Number of Sparse Graphs , 1995, Random Struct. Algorithms.

[127]  Jeff Kahn,et al.  Asymptotically Good List-Colorings , 1996, J. Comb. Theory A.

[128]  Roland Häggkvist,et al.  New Bounds on the List-Chromatic Index of the Complete Graph and Other Simple Graphs , 1997, Combinatorics, Probability and Computing.

[129]  Dhruv Mubayi,et al.  List coloring triangle‐free hypergraphs , 2013, Random Struct. Algorithms.

[130]  Daniela Kühn,et al.  Optimal path and cycle decompositions of dense quasirandom graphs , 2015, Electron. Notes Discret. Math..

[131]  James B. Shearer,et al.  A note on the independence number of triangle-free graphs , 1983, Discret. Math..

[132]  János Komlós,et al.  Extremal Uncrowded Hypergraphs , 1982, J. Comb. Theory, Ser. A.

[133]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[134]  Béla Bollobás,et al.  The independence ratio of regular graphs , 1981 .

[135]  Nicholas J. Cavenagh,et al.  On the chromatic index of Latin squares , 2015, Contributions Discret. Math..

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

[137]  Stefan Ehard,et al.  Pseudorandom hypergraph matchings , 2019, Combinatorics, Probability and Computing.