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[[Image:Graph star coloring.svg|thumb|right|300px|The star chromatic number of [[Dyck graph]] is 4, while the chromatic number is 2.]]
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In [[graph theory|graph-theoretic]] [[mathematics]], a '''star coloring''' of a graph ''G'' is a [[Graph coloring|(proper) vertex coloring]] in which every [[Path (graph theory)|path on four vertices]] uses at least three distinct colors. Equivalently, in a star coloring, the [[induced subgraph]]s formed by the vertices of any two colors has [[connected component (graph theory)|connected component]]s that are [[star graph]]s. Star coloring has been introduced by {{harvtxt|Grünbaum|1973}}.
The '''star chromatic number''' <math>\chi_s(G)</math> of ''G'' is the least number of colors needed to star color ''G''.
 
One generalization of star coloring is the closely related concept of [[acyclic coloring]], where it is required that every cycle uses at least three colors, so the two-color induced subgraphs are [[forest (graph theory)|forests]]. If we denote the acyclic chromatic number of a graph ''G'' by <math>\chi_a(G)</math>, we have that <math>\chi_a(G) \leq \chi_s(G)</math>, and in fact every star coloring of ''G'' is an acyclic coloring.
 
The star chromatic number has been proved to be bounded on every proper minor closed class by {{harvtxt|Nešetřil|Ossona de Mendez|2003}}. This results was further generalized by {{harvtxt|Nešetřil|Ossona de Mendez|2006}} to all low-tree-depth colorings (standard coloring and star coloring being low-tree-depth colorings with respective parameter 1 and 2).
 
== Complexity ==
It was demonstrated by {{harvtxt|Albertson|Chappell|Kierstead|Kündgen|2004}} that it is [[NP-complete]] to determine whether <math>\chi_s(G) \leq 3</math>, even when ''G'' is a graph that is both [[planar graph| planar]] and [[bipartite graph|bipartite]].
{{harvtxt|Coleman|Moré|1984}} showed that finding an optimal star coloring is [[NP-hard]] even when ''G'' is a bipartite graph.
 
== References ==
*{{citation
| last1 = Albertson | first1 = Michael O.
| last2 = Chappell | first2 = Glenn G.
| last3 = Kierstead | first3 = Hal A.
| last4 = Kündgen | first4 = André
| last5 = Ramamurthi | first5 = Radhika
| url = http://www.combinatorics.org/Volume_11/Abstracts/v11i1r26.html
| mr = 2056078
| title = Coloring with no 2-Colored ''P''<sub>4</sub>'s
| journal = The Electronic Journal of Combinatorics
| volume = 11 | issue = 1
| year = 2004}}.
*{{citation
| last1 = Coleman | first1 = Thomas F.
| last2 = Moré | first2 = Jorge
| mr = 0736293
| title = Estimation of sparse Hessian matrices and graph coloring problems
| journal = Mathematical Programming
| volume = 28 | issue = 3 | pages = 243–270
| year = 1984
| doi = 10.1007/BF02612334}}.
*{{citation
| last1 = Fertin | first1 = Guillaume
| last2 = Raspaud | first2 = André
| last3 = Reed | first3 = Bruce | author3-link = Bruce Reed (mathematician)
| journal = Journal of Graph Theory
| title = Star coloring of graphs
| volume = 47 | issue = 3 | pages = 163–182
| year = 2004
| mr = 2089462
| doi = 10.1002/jgt.20029}}.
*{{citation
| last = Grünbaum | first = Branko | authorlink=Branko Grünbaum
| doi = 10.1007/BF02764716 | mr = 0317982
| journal = Israel Journal of Mathematics
| title = Acyclic colorings of planar graphs
| volume = 14 | pages = 390–408
| year = 1973}}.
* {{citation
  | first1 = Jaroslav | last1 = Nešetřil | authorlink1=Jaroslav  Nešetřil
  | first2 = Patrice | last2 = Ossona de Mendez
  | mr = 2038495
  | series = Algorithms & Combinatorics | publisher = Springer-Verlag
  | volume = 25 | pages = 651–664
  | contribution = Colorings and homomorphisms of minor closed classes
  | title = Discrete & Computational Geometry: The Goodman-Pollack Festschrift
  | year = 2003}}.
* {{citation
  | first1 = Jaroslav | last1 = Nešetřil | authorlink1=Jaroslav  Nešetřil
  | first2 = Patrice | last2 = Ossona de Mendez
  | doi = 10.1016/j.ejc.2005.01.010 | mr = 2226435
  | journal = [[European Journal of Combinatorics]]
  | volume = 27 | issue = 6 | pages = 1022–1041
  | title = Tree depth, subgraph coloring and homomorphism bounds
  | year = 2006}}.
 
== External links ==
* [http://www.math.uiuc.edu/~west/regs/starcol.html Star colorings and acyclic colorings (1973)], present at the [http://www.math.uiuc.edu/~west/regs/ Research Experiences for Graduate Students (REGS)] at the University of Illinois, 2008.
 
[[Category:Graph coloring]]

Revision as of 23:30, 9 February 2014

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