### Abstract

A proper 2-tone k-coloring of a graph is a labeling of the vertices with elements from (formula presented.) such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph G, denoted τ _{2}(G) is the smallest k such that G admits a proper 2-tone k coloring. In this paper, we prove that w.h.p. for (formula presented.) where X represents the ordinary chromatic number. For sparse random graphs with p = c/n, c constant, we prove that (formula presented.) where Δ represents the maximum degree. For the more general concept of t-tone coloring, we achieve similar results.

Original language | English |
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Pages (from-to) | 1073-1086 |

Number of pages | 14 |

Journal | Graphs and Combinatorics |

Volume | 30 |

Issue number | 5 |

DOIs | |

State | Published - 1 Jan 2014 |

### Fingerprint

### Keywords

- Random graphs
- Tone colorings
- Vertex labelings

### Cite this

*Graphs and Combinatorics*,

*30*(5), 1073-1086. https://doi.org/10.1007/s00373-013-1341-9

}

*Graphs and Combinatorics*, vol. 30, no. 5, pp. 1073-1086. https://doi.org/10.1007/s00373-013-1341-9

**The t-Tone Chromatic Number of Random Graphs.** / Bal, Deepak; Bennett, Patrick; Dudek, Andrzej; Frieze, Alan.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - The t-Tone Chromatic Number of Random Graphs

AU - Bal, Deepak

AU - Bennett, Patrick

AU - Dudek, Andrzej

AU - Frieze, Alan

PY - 2014/1/1

Y1 - 2014/1/1

N2 - A proper 2-tone k-coloring of a graph is a labeling of the vertices with elements from (formula presented.) such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph G, denoted τ 2(G) is the smallest k such that G admits a proper 2-tone k coloring. In this paper, we prove that w.h.p. for (formula presented.) where X represents the ordinary chromatic number. For sparse random graphs with p = c/n, c constant, we prove that (formula presented.) where Δ represents the maximum degree. For the more general concept of t-tone coloring, we achieve similar results.

AB - A proper 2-tone k-coloring of a graph is a labeling of the vertices with elements from (formula presented.) such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph G, denoted τ 2(G) is the smallest k such that G admits a proper 2-tone k coloring. In this paper, we prove that w.h.p. for (formula presented.) where X represents the ordinary chromatic number. For sparse random graphs with p = c/n, c constant, we prove that (formula presented.) where Δ represents the maximum degree. For the more general concept of t-tone coloring, we achieve similar results.

KW - Random graphs

KW - Tone colorings

KW - Vertex labelings

UR - http://www.scopus.com/inward/record.url?scp=84906345851&partnerID=8YFLogxK

U2 - 10.1007/s00373-013-1341-9

DO - 10.1007/s00373-013-1341-9

M3 - Article

VL - 30

SP - 1073

EP - 1086

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 5

ER -