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Cordial labeling of hypertrees. (English) Zbl 1281.05118
Summary: Let $$H=(V,E)$$ be a hypergraph with vertex set $$V=\{v_1,v_2,\dots ,v_n\}$$ and edge set $$E=\{e_1,e_2,\dots ,e_m\}$$. A vertex labeling $$c:V\to\mathbb N$$ induces an edge labeling $$c^\ast :E\to\mathbb N$$ by the rule $$c^\ast (e_i)=\sum_{v_j\in e_i}c(v_j)$$. For integers $$k\geq 2$$ we study the existence of labelings satisfying the following condition: every residue class modulo $$k$$ occurs exactly $$\lfloor n/k\rfloor$$ or $$\lceil n/k\rceil$$ times in the sequence $$c(v_1),c(v_2),\dots ,c(v_n)$$ and exactly $$\lfloor m/k\rfloor$$ or $$\lceil m/k\rceil$$ times in the sequence $$c^\ast (e_1),c^\ast (e_2),\dots ,c^\ast (e_m)$$. Hypergraph $$H$$ is called $$k$$-cordial if it admits a labeling with these properties.
M. Hovey [ibid. 93, No. 2–3, 183–194 (1991; Zbl 0753.05059)] 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 I. Cahit’s theorem [Ars Comb. 23, 201–207 (1987; Zbl 0616.05056)], which states that every tree graph is 2-cordial. We also prove that every uniform hyperpath is $$k$$-cordial for every $$k$$.

##### MSC:
 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C65 Hypergraphs 05C05 Trees 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
##### Keywords:
$$k$$-cordial graph; hypergraph; hypergraph labeling; hypertree
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##### References:
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