# List coloring > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/List_coloring > Markdown URL: https://mediated.wiki/source/List_coloring.md > Source: https://en.wikipedia.org/wiki/List_coloring > Source revision: 1327743713 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{short description|Graph coloring where each vertex has a list of allowed colors}} In [graph theory](/source/graph_theory), a branch of [mathematics](/source/mathematics), '''list coloring''' is a type of [graph coloring](/source/graph_coloring) where each [vertex](/source/Vertex_(graph_theory)) can be restricted to a list of allowed colors. It was first studied in the 1970s in independent papers by [Vizing](/source/Vadim_G._Vizing) and by [Erdős](/source/Paul_Erd%C5%91s), [Rubin](/source/Arthur_Rubin), and Taylor.{{citation|last1=Jensen|first1=Tommy R.|last2=Toft|first2=Bjarne|year=1995|title=Graph coloring problems|location=New York|publisher=Wiley-Interscience|isbn=0-471-02865-7|contribution=1.9 List coloring|pages=18–21}} ==Definition== Given a graph {{mvar|G}} and given a set {{math|''L''(''v'')}} of colors for each vertex {{mvar|v}} (called a '''list'''), a '''list coloring''' is a ''choice function'' that maps every vertex {{mvar|v}} to a color in the list {{math|''L''(''v'')}}. As with graph coloring, a list coloring is generally assumed to be '''proper''', meaning no two [adjacent vertices](/source/Adjacent_vertex) receive the same color. A graph is '''{{mvar|k}}-choosable''' (or '''{{mvar|k}}-list-colorable''') if it has a proper list coloring no matter how one assigns a list of {{mvar|k}} colors to each vertex. The '''choosability''' (or '''list colorability''' or '''list chromatic number''') {{math|ch(''G'')}} of a graph {{mvar|G}} is the least number {{mvar|k}} such that {{mvar|G}} is {{mvar|k}}-choosable. More generally, for a function {{mvar|f}} assigning a positive integer {{math|''f''(''v'')}} to each vertex {{mvar|v}}, a graph {{mvar|G}} is '''{{mvar|f}}-choosable''' (or '''{{mvar|f}}-list-colorable''') if it has a list coloring no matter how one assigns a list of {{math|''f''(''v'')}} colors to each vertex {{mvar|v}}. In particular, if {{math|1=''f''(''v'') = ''k''}} for all vertices {{mvar|v}}, {{mvar|f}}-choosability corresponds to {{mvar|k}}-choosability. ==Examples== Consider the complete [bipartite graph](/source/bipartite_graph) {{math|1=''G'' = ''K''2,4}}, having six vertices {{mvar|A}}, {{mvar|B}}, {{mvar|W}}, {{mvar|X}}, {{mvar|Y}}, {{mvar|Z}} such that {{mvar|A}} and {{mvar|B}} are each connected to all of {{mvar|W}}, {{mvar|X}}, {{mvar|Y}}, and {{mvar|Z}}, and no other vertices are connected. As a bipartite graph, {{mvar|G}} has usual chromatic number 2: one may color {{mvar|A}} and {{mvar|B}} in one color and {{mvar|W}}, {{mvar|X}}, {{mvar|Y}}, {{mvar|Z}} in another and no two adjacent vertices will have the same color. On the other hand, {{mvar|G}} has list-chromatic number larger than 2, as the following construction shows: assign to {{mvar|A}} and {{mvar|B}} the lists {red, blue} and {green, black}. Assign to the other four vertices the lists {red, green}, {red, black}, {blue, green}, and {blue, black}. No matter which choice one makes of a color from the list of {{mvar|A}} and a color from the list of {{mvar|B}}, there will be some other vertex such that both of its choices are already used to color its neighbors. Thus, {{mvar|G}} is not 2-choosable. On the other hand, it is easy to see that {{mvar|G}} is 3-choosable: picking arbitrary colors for the vertices {{mvar|A}} and {{mvar|B}} leaves at least one available color for each of the remaining vertices, and these colors may be chosen arbitrarily. [[File:List-coloring-K-3-27.svg|thumb|300px|A list coloring instance on the [complete bipartite graph](/source/complete_bipartite_graph) {{math|''K''3,27}} with three colors per vertex. No matter which colors are chosen for the three central vertices, one of the outer 27 vertices will be uncolorable, showing that the list chromatic number of {{math|''K''3,27}} is at least four.]] More generally, let {{mvar|q}} be a positive integer, and let {{mvar|G}} be the [complete bipartite graph](/source/complete_bipartite_graph) {{mvar|Kq,qq}}. Let the available colors be represented by the {{math|''q''2}} different two-digit numbers in [radix](/source/radix) {{mvar|q}}. On one side of the bipartition, let the {{mvar|q}} vertices be given sets of colors {{math|{''i''0, ''i''1, ''i''2, ...}}} in which the first digits are equal to each other, for each of the {{mvar|q}} possible choices of the first digit {{mvar|i}}. On the other side of the bipartition, let the {{mvar|qq}} vertices be given sets of colors {{math|{0''a'', 1''b'', 2''c'', ...}}} in which the first digits are all distinct, for each of the {{mvar|qq}} possible choices of the {{mvar|q}}-tuple {{math|(''a'', ''b'', ''c'', ...).}} The illustration shows a larger example of the same construction, with {{math|1=''q'' = 3}}. Then, {{mvar|G}} does not have a list coloring for {{mvar|L}}: no matter what set of colors is chosen for the vertices on the small side of the bipartition, this choice will conflict with all of the colors for one of the vertices on the other side of the bipartition. For instance if the vertex with color set {00,01} is colored 01, and the vertex with color set {10,11} is colored 10, then the vertex with color set {01,10} cannot be colored. Therefore, the list chromatic number of {{mvar|G}} is at least {{math|''q'' + 1}}.{{citation | last = Gravier | first = Sylvain | doi = 10.1016/0012-365X(95)00350-6 | issue = 1–3 | journal = [Discrete Mathematics](/source/Discrete_Mathematics_(journal)) | mr = 1388650 | pages = 299–302 | title = A Hajós-like theorem for list coloring | volume = 152 | year = 1996| doi-access = free }}. Similarly, if n=\tbinom{2k-1}{k}, then the complete bipartite graph {{mvar|Kn,n}} is not {{mvar|k}}-choosable. For, suppose that {{math|2''k'' − 1}} colors are available in total, and that, on a single side of the bipartition, each vertex has available to it a different {{mvar|k}}-tuple of these colors than each other vertex. Then, each side of the bipartition must use at least {{mvar|k}} colors, because every set of {{math|k − 1}} colors will be disjoint from the list of one vertex. Since at least {{mvar|k}} colors are used on one side and at least {{mvar|k}} are used on the other, there must be one color which is used on both sides, but this implies that two adjacent vertices have the same color. In particular, the [utility graph](/source/utility_graph) {{math|''K''3,3}} has list-chromatic number at least three, and the graph {{math|''K''10,10}} has list-chromatic number at least four.{{citation|last1=Erdős|first1=P.|author1-link=Paul Erdős|last2=Rubin|first2=A. L.|author2-link=Arthur Rubin|last3=Taylor|first3=H.|year=1979|url=http://www.math-inst.hu/~p_erdos/1980-07.pdf|contribution=Choosability in graphs|title=Proc. West Coast Conference on Combinatorics, Graph Theory and Computing, Arcata|series=Congressus Numerantium|volume=26|pages=125–157|access-date=2008-11-10|archive-url=https://web.archive.org/web/20160309235325/http://www.math-inst.hu/~p_erdos/1980-07.pdf|archive-date=2016-03-09|url-status=dead}} ==Properties== For a graph {{mvar|G}}, let {{math|''χ''(''G'')}} denote the [chromatic number](/source/Graph_coloring) and {{math|Δ(''G'')}} the [maximum degree](/source/Glossary_of_graph_theory) of {{mvar|G}}. The list coloring number {{math|ch(''G'')}} satisfies the following properties. # {{math|ch(''G'') ≥ ''χ''(''G'')}}. A {{mvar|k}}-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of {{mvar|k}} colors, which corresponds to a usual {{mvar|k}}-coloring. # {{math|ch(''G'')}} cannot be bounded in terms of chromatic number in general, that is, there is no function {{mvar|f}} such that {{math|ch(''G'') ≤ ''f''(''χ''(''G''))}} holds for every graph {{mvar|G}}. In particular, as the complete bipartite graph examples show, there exist graphs with {{math|1=''χ''(''G'') = 2}} but with {{math|ch(''G'')}} arbitrarily large. # {{math|ch(''G'') ≤ ''χ''(''G'') ln(''n'')}} where {{mvar|n}} is the number of vertices of {{mvar|G}}.{{Citation | last = Eaton | first = Nancy | title = On two short proofs about list coloring - Part 1 | work = Talk | year = 2003 |url=http://www.math.uri.edu/~eaton/TalkUriOct03P1.pdf | accessdate = May 29, 2010 | archive-url=https://web.archive.org/web/20170829220122/http://www.math.uri.edu/~eaton/TalkUriOct03P1.pdf | archive-date = August 29, 2017 | url-status = dead }}{{Citation | last = Eaton | first = Nancy | title = On two short proofs about list coloring - Part 2 | work = Talk | year = 2003 |url=http://www.math.uri.edu/~eaton/TalkUriOct03P2.pdf | accessdate = May 29, 2010 | archive-url=https://web.archive.org/web/20170830012324/http://www.math.uri.edu/~eaton/TalkUriOct03P2.pdf | archive-date = August 30, 2017 | url-status = dead }} # {{math|ch(''G'') ≤ Δ(''G'') + 1}}.{{citation|authorlink=Vadim G. Vizing|last=Vizing|first=V. G.|year=1976|title=Vertex colorings with given colors|language=Russian|journal=Metody Diskret. Analiz.|volume=29|pages=3–10}} # {{math|ch(''G'') ≤ 5}} if {{mvar|G}} is a [planar graph](/source/planar_graph).{{citation|last=Thomassen|first=Carsten|authorlink=Carsten Thomassen (mathematician)|year=1994|title=Every planar graph is 5-choosable|journal=Journal of Combinatorial Theory, Series B|volume=62|pages=180–181|doi=10.1006/jctb.1994.1062|doi-access=free}} # {{math|ch(''G'') ≤ 3}} if {{mvar|G}} is a [bipartite](/source/bipartite_graph) planar graph.{{citation|doi=10.1007/BF01204715|last1=Alon|first1=Noga|author1-link=Noga Alon|last2=Tarsi|first2=Michael|year=1992|title=Colorings and orientations of graphs|journal=Combinatorica|volume=12|issue=2|pages=125–134|s2cid=45528500 |citeseerx=10.1.1.106.9928}} ==Computing choosability and (''a'', ''b'')-choosability== Two algorithmic problems have been considered in the literature: # {{mvar|k}}-''choosability'': decide whether a given graph is {{mvar|k}}-choosable for a given {{mvar|k}}, and # {{math|(''a'', ''b'')}}-''choosability'': decide whether a given graph is {{mvar|f}}-choosable for a given function f : V \to \{a,\dots,b\}. It is known that {{mvar|k}}-choosability in bipartite graphs is \Pi^p_2-complete for any {{math|''k'' ≥ 3}}, and the same applies for 4-choosability in planar graphs, 3-choosability in planar [triangle-free graph](/source/triangle-free_graph)s, and (2, 3)-choosability in [bipartite](/source/bipartite_graph) planar graphs.{{citation|doi=10.1016/0012-365X(95)00104-5|last=Gutner|first=Shai|year=1996|arxiv=0802.2668 |title=The complexity of planar graph choosability|journal=[Discrete Mathematics](/source/Discrete_Mathematics_(journal))|volume=159|issue=1|pages=119–130|s2cid=1392057 }}.{{citation|last1=Gutner|first1=Shai|last2=Tarsi|first2=Michael|year=2009|doi=10.1016/j.disc.2008.04.061|title=Some results on (''a'':''b'')-choosability|journal=[Discrete Mathematics](/source/Discrete_Mathematics_(journal))|volume=309|issue=8|pages=2260–2270|doi-access=}} For {{math|P5}}-free graphs, that is, graphs [excluding](/source/forbidden_graph_characterization) a 5-vertex [path graph](/source/path_graph), {{mvar|k}}-choosability is [fixed-parameter tractable](/source/fixed-parameter_tractable). {{citation | last1 = Heggernes | first1 = Pinar | author1-link = Pinar Heggernes | last2 = Golovach | first2 = Petr | contribution = Choosability of P5-free graphs | doi = | pages = 382–391 | publisher = Springer-Verlag | series = Lecture Notes on Computer Science | title = Mathematical Foundations of Computer Science | contribution-url=http://www.ii.uib.no/~pinar/Choosability.pdf | volume = 5734 | year = 2009}} It is possible to test whether a graph is 2-choosable in [linear time](/source/linear_time) by repeatedly deleting vertices of degree zero or one until reaching the [2-core](/source/Degeneracy_(graph_theory)) of the graph, after which no more such deletions are possible. The initial graph is 2-choosable [if and only if](/source/if_and_only_if) its 2-core is either an even cycle or a [theta graph](/source/theta_graph) formed by three paths with shared endpoints, with two paths of length two and the third path having any even length. ==Applications== List coloring arises in practical problems concerning channel/frequency assignment.{{citation | last1 = Wang | first1 = Wei | last2 = Liu | first2 = Xin | contribution = List-coloring based channel allocation for open-spectrum wireless networks | doi = 10.1109/VETECF.2005.1558001 | pages = 690–694 | title = 2005 IEEE 62nd Vehicular Technology Conference (VTC 2005-Fall) | volume = 1 | year = 2005| isbn = 0-7803-9152-7 | s2cid = 14952297 }}.{{citation | last1 = Garg | first1 = N. | last2 = Papatriantafilou | first2 = M. | last3 = Tsigas | first3 = P. | contribution = Distributed list coloring: how to dynamically allocate frequencies to mobile base stations | doi = 10.1109/SPDP.1996.570312 | pages = 18–25 | title = Eighth IEEE Symposium on Parallel and Distributed Processing | year = 1996| hdl = 21.11116/0000-0001-1AE6-F | isbn = 0-8186-7683-3 | s2cid = 3319306 | hdl-access = free }}. == See also == {{Wiktionary|choosability}} * [List edge-coloring](/source/List_edge-coloring) == References == '''Further reading''' *{{Citation |author1=Aigner, Martin |author2=Ziegler, Günter | title=Proofs from THE BOOK | publisher=Springer-Verlag | location=Berlin, New York | year=2009 | edition=4th | isbn=978-3-642-00855-9}}, Chapter 34 ''Five-coloring plane graphs''. *Diestel, Reinhard. ''Graph Theory''. 3rd edition, Springer, 2005. Chapter 5.4 ''List Colouring''. [http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/GraphTheoryIII.pdf electronic edition available for download] {{DEFAULTSORT:List Coloring}} Category:Graph coloring --- Adapted from the Wikipedia article [List coloring](https://en.wikipedia.org/wiki/List_coloring) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/List_coloring?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.