### Abstract

An edge-colored graph G is k-proper connected if every pair of vertices is connected by k internally pairwise vertex-disjoint proper colored paths. The k-proper connection number of a connected graph G, denoted by p ^{ck}(G), is the smallest number of colors that are needed to color the edges of G in order to make it k-proper connected. In this paper we prove several upper bounds for p ^{ck}(G). We state some conjectures for general and bipartite graphs, and we prove them for the case when k=1. In particular, we prove a variety of conditions on G which imply p ^{c1}(G)=2.

Original language | English |
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Pages (from-to) | 2550-2560 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 312 |

Issue number | 17 |

DOIs | |

Publication status | Published - Sep 6 2012 |

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### Keywords

- Proper coloring
- Proper connection

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*312*(17), 2550-2560. https://doi.org/10.1016/j.disc.2011.09.003

**Proper connection of graphs.** / Borozan, Valentin; Fujita, Shinya; Gerek, Aydin; Magnant, Colton; Manoussakis, Yannis; Montero, Leandro; Tuza, Z.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 312, no. 17, pp. 2550-2560. https://doi.org/10.1016/j.disc.2011.09.003

}

TY - JOUR

T1 - Proper connection of graphs

AU - Borozan, Valentin

AU - Fujita, Shinya

AU - Gerek, Aydin

AU - Magnant, Colton

AU - Manoussakis, Yannis

AU - Montero, Leandro

AU - Tuza, Z.

PY - 2012/9/6

Y1 - 2012/9/6

N2 - An edge-colored graph G is k-proper connected if every pair of vertices is connected by k internally pairwise vertex-disjoint proper colored paths. The k-proper connection number of a connected graph G, denoted by p ck(G), is the smallest number of colors that are needed to color the edges of G in order to make it k-proper connected. In this paper we prove several upper bounds for p ck(G). We state some conjectures for general and bipartite graphs, and we prove them for the case when k=1. In particular, we prove a variety of conditions on G which imply p c1(G)=2.

AB - An edge-colored graph G is k-proper connected if every pair of vertices is connected by k internally pairwise vertex-disjoint proper colored paths. The k-proper connection number of a connected graph G, denoted by p ck(G), is the smallest number of colors that are needed to color the edges of G in order to make it k-proper connected. In this paper we prove several upper bounds for p ck(G). We state some conjectures for general and bipartite graphs, and we prove them for the case when k=1. In particular, we prove a variety of conditions on G which imply p c1(G)=2.

KW - Proper coloring

KW - Proper connection

UR - http://www.scopus.com/inward/record.url?scp=84862667323&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862667323&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2011.09.003

DO - 10.1016/j.disc.2011.09.003

M3 - Article

AN - SCOPUS:84862667323

VL - 312

SP - 2550

EP - 2560

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 17

ER -