### Abstract

A colored graph is a graph whose vertices have been properly, though not necessarily optimally colored, with integers. Colored graphs have a natural orientation in which edges are directed from the end point with smaller color to the end point with larger color. A subgraph of a colored graph is colorful if each of its vertices has a distinct color. We prove that there exists a function f(k, n) such that for any colored graph G, if _{χ}(G)>f(ω(G), n) then G induces either a colorful out directed star with n leaves or a colorful directed path on n vertices. We also show that this result would be false if either alternative was omitted. Our results provide a solution to Problem 115, Discrete Math. 79.

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

Pages (from-to) | 165-169 |

Number of pages | 5 |

Journal | Discrete Mathematics |

Volume | 101 |

Issue number | 1-3 |

DOIs | |

State | Published - May 29 1992 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*101*(1-3), 165-169. https://doi.org/10.1016/0012-365X(92)90600-K

**Colorful induced subgraphs.** / Kierstead, Henry; Trotter, W. T.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 101, no. 1-3, pp. 165-169. https://doi.org/10.1016/0012-365X(92)90600-K

}

TY - JOUR

T1 - Colorful induced subgraphs

AU - Kierstead, Henry

AU - Trotter, W. T.

PY - 1992/5/29

Y1 - 1992/5/29

N2 - A colored graph is a graph whose vertices have been properly, though not necessarily optimally colored, with integers. Colored graphs have a natural orientation in which edges are directed from the end point with smaller color to the end point with larger color. A subgraph of a colored graph is colorful if each of its vertices has a distinct color. We prove that there exists a function f(k, n) such that for any colored graph G, if χ(G)>f(ω(G), n) then G induces either a colorful out directed star with n leaves or a colorful directed path on n vertices. We also show that this result would be false if either alternative was omitted. Our results provide a solution to Problem 115, Discrete Math. 79.

AB - A colored graph is a graph whose vertices have been properly, though not necessarily optimally colored, with integers. Colored graphs have a natural orientation in which edges are directed from the end point with smaller color to the end point with larger color. A subgraph of a colored graph is colorful if each of its vertices has a distinct color. We prove that there exists a function f(k, n) such that for any colored graph G, if χ(G)>f(ω(G), n) then G induces either a colorful out directed star with n leaves or a colorful directed path on n vertices. We also show that this result would be false if either alternative was omitted. Our results provide a solution to Problem 115, Discrete Math. 79.

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UR - http://www.scopus.com/inward/citedby.url?scp=38249013388&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(92)90600-K

DO - 10.1016/0012-365X(92)90600-K

M3 - Article

AN - SCOPUS:38249013388

VL - 101

SP - 165

EP - 169

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -