### Abstract

A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic k-coloring can be refined to a star coloring with at most (2k ^{2} - k) colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are 7-colorable, and planar graphs of girth at least 7 are 9-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width t can be star colored with ( _{2} ^{t+2}) colors, and we show that this is best possible.

Original language | English (US) |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 11 |

Issue number | 1 R |

State | Published - Mar 31 2004 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

_{4}'s.

*Electronic Journal of Combinatorics*,

*11*(1 R).

**Coloring with no 2-colored P _{4}'s.** / Albertson, Michael O.; Chappell, Glenn G.; Kierstead, Henry; Kündgen, André; Ramamurthi, Radhika.

Research output: Contribution to journal › Article

_{4}'s',

*Electronic Journal of Combinatorics*, vol. 11, no. 1 R.

_{4}'s. Electronic Journal of Combinatorics. 2004 Mar 31;11(1 R).

}

TY - JOUR

T1 - Coloring with no 2-colored P 4's

AU - Albertson, Michael O.

AU - Chappell, Glenn G.

AU - Kierstead, Henry

AU - Kündgen, André

AU - Ramamurthi, Radhika

PY - 2004/3/31

Y1 - 2004/3/31

N2 - A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic k-coloring can be refined to a star coloring with at most (2k 2 - k) colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are 7-colorable, and planar graphs of girth at least 7 are 9-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width t can be star colored with ( 2 t+2) colors, and we show that this is best possible.

AB - A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic k-coloring can be refined to a star coloring with at most (2k 2 - k) colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are 7-colorable, and planar graphs of girth at least 7 are 9-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width t can be star colored with ( 2 t+2) colors, and we show that this is best possible.

UR - http://www.scopus.com/inward/record.url?scp=84862383319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862383319&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84862383319

VL - 11

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1 R

ER -