### Abstract

Let H=(V,E) be a hypergraph with vertex set V={^{v1}, ^{v2},⋯,^{vn}} and edge set E={^{e1}, ^{e2},⋯,^{em}}. A vertex labeling c:V→N induces an edge labeling ^{c*}:E→N by the rule ^{c*}( ^{ei})=Σ_{vjâ̂̂ ei}c(^{vj}). For integers k≥2 we study the existence of labelings satisfying the following condition: every residue class modulo k occurs exactly ⌊n/k⌋ or ⌊n/k⌋ times in the sequence c(^{v1}),c(^{v2}),⋯,c(^{vn}) and exactly ⌊m/k⌋ or ⌊m/k⌋ times in the sequence ^{c*}(^{e1}),^{c*}(^{e2}), ⋯^{c*}(^{em}). Hypergraph H is called k-cordial if it admits a labeling with these properties. Hovey [M. Hovey, A-cordial graphs, Discrete Math. 93 (1991) 183-194] raised the conjecture (still open for k>5) that if H is a tree graph, then it is k-cordial for every k. Here we investigate the analogous problem for hypertrees (connected hypergraphs without cycles) and present various sufficient conditions on H to be k-cordial. From our theorems it follows that every k-uniform hypertree is k-cordial, and every hypertree with n or m odd is 2-cordial. Both of these results generalize Cahit's theorem [I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combin. 23 (1987) 201-207] which states that every tree graph is 2-cordial. We also prove that every uniform hyperpath is k-cordial for every k.

Original language | English |
---|---|

Pages (from-to) | 2518-2524 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 313 |

Issue number | 22 |

DOIs | |

Publication status | Published - 2013 |

### Fingerprint

### Keywords

- Hypergraph
- Hypergraph labeling
- Hypertree
- k-cordial graph

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*313*(22), 2518-2524. https://doi.org/10.1016/j.disc.2013.07.025

**Cordial labeling of hypertrees.** / Cichacz, Sylwia; Görlich, Agnieszka; Tuza, Z.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 313, no. 22, pp. 2518-2524. https://doi.org/10.1016/j.disc.2013.07.025

}

TY - JOUR

T1 - Cordial labeling of hypertrees

AU - Cichacz, Sylwia

AU - Görlich, Agnieszka

AU - Tuza, Z.

PY - 2013

Y1 - 2013

N2 - Let H=(V,E) be a hypergraph with vertex set V={v1, v2,⋯,vn} and edge set E={e1, e2,⋯,em}. A vertex labeling c:V→N induces an edge labeling c*:E→N by the rule c*( ei)=Σvjâ̂̂ eic(vj). For integers k≥2 we study the existence of labelings satisfying the following condition: every residue class modulo k occurs exactly ⌊n/k⌋ or ⌊n/k⌋ times in the sequence c(v1),c(v2),⋯,c(vn) and exactly ⌊m/k⌋ or ⌊m/k⌋ times in the sequence c*(e1),c*(e2), ⋯c*(em). Hypergraph H is called k-cordial if it admits a labeling with these properties. Hovey [M. Hovey, A-cordial graphs, Discrete Math. 93 (1991) 183-194] raised the conjecture (still open for k>5) that if H is a tree graph, then it is k-cordial for every k. Here we investigate the analogous problem for hypertrees (connected hypergraphs without cycles) and present various sufficient conditions on H to be k-cordial. From our theorems it follows that every k-uniform hypertree is k-cordial, and every hypertree with n or m odd is 2-cordial. Both of these results generalize Cahit's theorem [I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combin. 23 (1987) 201-207] which states that every tree graph is 2-cordial. We also prove that every uniform hyperpath is k-cordial for every k.

AB - Let H=(V,E) be a hypergraph with vertex set V={v1, v2,⋯,vn} and edge set E={e1, e2,⋯,em}. A vertex labeling c:V→N induces an edge labeling c*:E→N by the rule c*( ei)=Σvjâ̂̂ eic(vj). For integers k≥2 we study the existence of labelings satisfying the following condition: every residue class modulo k occurs exactly ⌊n/k⌋ or ⌊n/k⌋ times in the sequence c(v1),c(v2),⋯,c(vn) and exactly ⌊m/k⌋ or ⌊m/k⌋ times in the sequence c*(e1),c*(e2), ⋯c*(em). Hypergraph H is called k-cordial if it admits a labeling with these properties. Hovey [M. Hovey, A-cordial graphs, Discrete Math. 93 (1991) 183-194] raised the conjecture (still open for k>5) that if H is a tree graph, then it is k-cordial for every k. Here we investigate the analogous problem for hypertrees (connected hypergraphs without cycles) and present various sufficient conditions on H to be k-cordial. From our theorems it follows that every k-uniform hypertree is k-cordial, and every hypertree with n or m odd is 2-cordial. Both of these results generalize Cahit's theorem [I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combin. 23 (1987) 201-207] which states that every tree graph is 2-cordial. We also prove that every uniform hyperpath is k-cordial for every k.

KW - Hypergraph

KW - Hypergraph labeling

KW - Hypertree

KW - k-cordial graph

UR - http://www.scopus.com/inward/record.url?scp=84883024841&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883024841&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2013.07.025

DO - 10.1016/j.disc.2013.07.025

M3 - Article

VL - 313

SP - 2518

EP - 2524

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 22

ER -