Cordial labeling of hypertrees

Sylwia Cichacz, Agnieszka Görlich, Z. Tuza

Research output: Contribution to journalArticle

2 Citations (Scopus)

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)=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.

Original languageEnglish
Pages (from-to)2518-2524
Number of pages7
JournalDiscrete Mathematics
Volume313
Issue number22
DOIs
Publication statusPublished - 2013

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Hypertree
Labeling
Hypergraph
Graph in graph theory
Vertex Labeling
Edge Labeling
Theorem
Modulo
Odd
Cycle
Generalise
Integer
Sufficient Conditions
Vertex of a graph

Keywords

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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

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

In: Discrete Mathematics, Vol. 313, No. 22, 2013, p. 2518-2524.

Research output: Contribution to journalArticle

Cichacz, S, Görlich, A & Tuza, Z 2013, 'Cordial labeling of hypertrees', Discrete Mathematics, vol. 313, no. 22, pp. 2518-2524. https://doi.org/10.1016/j.disc.2013.07.025
Cichacz, Sylwia ; Görlich, Agnieszka ; Tuza, Z. / Cordial labeling of hypertrees. In: Discrete Mathematics. 2013 ; Vol. 313, No. 22. pp. 2518-2524.
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