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In [[Graph theory|graph theory]], a branch of [[Mathematics|mathematics]], '''list coloring''' is a type of [[Graph coloring|graph coloring]] where each vertex can be restricted to a list of allowed colors.  It was first studied in the 1970s in independent papers by [[Biography:Vadim G. Vizing|Vizing]]
and by [[Biography:Paul Erdős|Erdős]], Rubin, and Taylor.<ref>{{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}}</ref>

==Definition==

Given a graph ''G'' and given a set ''L''(''v'') of colors for each vertex ''v'' (called a '''list'''), a '''list coloring''' is a ''choice function'' that maps every vertex ''v'' to a color in the list ''L''(''v'').  As with graph coloring, a list coloring is generally assumed to be '''proper''', meaning no two adjacent vertices receive the same color.  A graph is '''''k''-choosable''' (or '''''k''-list-colorable''') if it has a proper list coloring no matter how one assigns a list of ''k'' colors to each vertex.  The '''choosability''' (or '''list colorability''' or '''list chromatic number''') ch(''G'') of a graph ''G'' is the least number ''k'' such that ''G'' is ''k''-choosable.

More generally, for a function ''f'' assigning a positive integer ''f''(''v'') to each vertex ''v'', a graph ''G'' is '''''f''-choosable''' (or '''''f''-list-colorable''') if it has a list coloring no matter how one assigns a list of ''f''(''v'') colors to each vertex ''v''.  In particular, if <math>f(v) = k</math> for all vertices ''v'', ''f''-choosability corresponds to ''k''-choosability.

==Examples==
Consider the complete [[Bipartite graph|bipartite graph]] ''G'' = ''K''<sub>2,4</sub>, having six vertices ''A'', ''B'', ''W'', ''X'', ''Y'', ''Z'' such that ''A'' and ''B'' are each connected to all of ''W'', ''X'', ''Y'', and ''Z'', and no other vertices are connected.  As a bipartite graph, ''G'' has usual chromatic number 2: one may color ''A'' and ''B'' in one color and ''W'', ''X'', ''Y'', ''Z'' in another and no two adjacent vertices will have the same color.  On the other hand, ''G'' has list-chromatic number larger than 2, as the following construction shows: assign to ''A'' and ''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 ''A'' and a color from the list of ''B'', there will be some other vertex such that both of its choices are already used to color its neighbors.  Thus, ''G'' is not 2-choosable.

On the other hand, it is easy to see that ''G'' is 3-choosable: picking arbitrary colors for the vertices ''A'' and ''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|complete bipartite graph]] ''K''<sub>3,27</sub> 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 ''K''<sub>3,27</sub> is at least four.]]
More generally, let ''q'' be a positive integer, and let ''G'' be the [[Complete bipartite graph|complete bipartite graph]] ''K''<sub>''q'',''q''<sup>''q''</sup></sub>. Let the available colors be represented by the ''q''<sup>2</sup> different two-digit numbers in [[Radix|radix]] ''q''.
On one side of the bipartition, let the ''q'' vertices be given sets of colors {{nowrap|{''i''0, ''i''1, ''i''2, ...}}} in which the first digits are equal to each other, for each of the ''q'' possible choices of the first digit&nbsp;''i''.
On the other side of the bipartition, let the ''q<sup>q</sup>'' vertices be given sets of colors {{nowrap|{0''a'', 1''b'', 2''c'', ...}}} in which the first digits are all distinct, for each of the ''q<sup>q</sup>'' possible choices of the ''q''-tuple {{nowrap|(''a'', ''b'', ''c'', ...).}}
The illustration shows a larger example of the same construction, with ''q''&nbsp;=&nbsp;3.

Then, ''G'' does not have a list coloring for ''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 ''G'' is at least ''q''&nbsp;+&nbsp;1.<ref name="g96">{{citation
 | last = Gravier | first = Sylvain
 | doi = 10.1016/0012-365X(95)00350-6
 | issue = 1–3
 | journal = Discrete Mathematics
 | mr = 1388650
 | pages = 299–302
 | title = A Hajós-like theorem for list coloring
 | volume = 152
 | year = 1996| doi-access = free
 }}.</ref>

Similarly, if <math>n=\binom{2k-1}{k}</math>, then the complete bipartite graph ''K''<sub>''n'', ''n''</sub> is not ''k''-choosable. For, suppose that 2''k''&nbsp;&minus;&nbsp;1 colors are available in total, and that, on a single side of the bipartition, each vertex has available to it a different ''k''-tuple of these colors than each other vertex. Then, each side of the bipartition must use at least ''k'' colors, because every set of ''k''&nbsp;&minus;&nbsp;1 colors will be disjoint from the list of one vertex. Since at least ''k'' colors are used on one side and at least ''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|utility graph]] ''K''<sub>3,3</sub> has list-chromatic number at least three, and the graph ''K''<sub>10,10</sub> has list-chromatic number at least four.<ref name="erdos">{{citation|last1=Erdős|first1=P.|last2=Rubin|first2=A. L.|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}}</ref>

==Properties==

For a graph ''G'', let χ(''G'') denote the [[Graph coloring|chromatic number]]  and Δ(''G'') the [[Glossary of graph theory|maximum degree]] of ''G''.  The list coloring number ch(''G'') satisfies the following properties.
# ch(''G'') ≥ χ(''G'').  A ''k''-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of ''k'' colors, which corresponds to a usual ''k''-coloring.
# ch(''G'') cannot be bounded in terms of chromatic number in general, that is, there is no function ''f'' such that ch(''G'') ≤ ''f''(χ(''G'')) holds for every graph ''G''.  In particular,  as the complete bipartite graph examples show, there exist graphs with χ(''G'')&nbsp;=&nbsp;2 but with ch(''G'') arbitrarily large.<ref name="g96"/>
# ch(''G'') ≤ χ(''G'') ln(''n'') where ''n'' is the number of vertices of ''G''.<ref>{{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
  }}</ref><ref>{{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
  }}</ref>
# ch(''G'') ≤ Δ(''G'') + 1.<ref name="erdos"/><ref name="vizing">{{citation|last=Vizing|first=V. G.|year=1976|title=Vertex colorings with given colors|language=Russian|journal=Metody Diskret. Analiz.|volume=29|pages=3–10}}</ref>
# ch(''G'') ≤ 5 if ''G'' is a [[Planar graph|planar graph]].<ref name="thomassen">{{citation|last=Thomassen|first=Carsten|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}}</ref>
# ch(''G'') ≤ 3 if ''G'' is a [[Bipartite graph|bipartite]] planar graph.<ref name="alon">{{citation|doi=10.1007/BF01204715|last1=Alon|first1=Noga|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}}</ref>

==Computing choosability and (''a'', ''b'')-choosability==

Two algorithmic problems have been considered in the literature:
# ''k''-''choosability'': decide whether a given graph is ''k''-choosable for a given ''k'', and
# (''a'', ''b'')-''choosability'': decide whether a given graph is ''f''-choosable for a given function <math>f : V \to \{a,\dots,b\}</math>.
It is known that ''k''-choosability in bipartite graphs is <math>\Pi^p_2</math>-complete for any ''k'' ≥ 3, and the same applies for 4-choosability in planar graphs, 3-choosability in planar [[Triangle-free graph|triangle-free graph]]s, and (2, 3)-choosability in [[Bipartite graph|bipartite]] planar graphs.<ref name="gutner">{{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|volume=159|issue=1|pages=119–130|s2cid=1392057 }}.</ref><ref name="GutnerTarsi">{{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|volume=309|issue=8|pages=2260–2270|doi-access=}}</ref> For P<sub>5</sub>-free graphs, that is, graphs [[Forbidden graph characterization|excluding]] a 5-vertex [[Path graph|path graph]], ''k''-choosability is fixed-parameter tractable.
<ref>{{citation
 | last2 = Golovach | first2 = Petr
 | contribution = Choosability of P<sub>5</sub>-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}}
</ref>

It is possible to test whether a graph is 2-choosable in linear time by repeatedly deleting vertices of degree zero or one until reaching the [[Degeneracy (graph theory)|2-core]] of the graph, after which no more such deletions are possible. The initial graph is 2-choosable if and only if its 2-core is either an even cycle or a [[Theta graph|theta graph]] formed by three paths with shared endpoints, with two paths of length two and the third path having any even length.<ref name="erdos"/>

==Applications==

List coloring arises in practical problems concerning channel/frequency assignment.<ref>{{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| s2cid = 14952297
 }}.</ref><ref>{{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
 | s2cid = 3319306
 | hdl-access = free
 }}.</ref>

== See also ==
* [[List edge-coloring]]

== References ==
<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''.<!--(sic! British English)--> [http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/GraphTheoryIII.pdf electronic edition available for download]

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[[Category:Graph coloring]]

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