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A-cordial graphs. (English) Zbl 0753.05059
Let $$A$$ be an Abelian group. A labelling of a graph $$G=(V,E)$$ is any mapping $$f:V\to A$$. Every such mapping induces a mapping $$f_ E:E\to A$$ by $$f_ E(xy):=f(x)+f(y)$$ for $$xy\in E$$. A graph is called $$A$$-cordial if there is some labelling $$f$$ such that $$\| f^{-1}(a)| -| f^{-1}(b)\|\leq 1$$ and $$\| f_ E^{-1}(a)| -| f_ E^{-1}(b)\|\leq 1$$ for all $$a,b\in A$$. This is a generalization of the notions of harmonious
(i.e. $${\mathcal Z}_{| E|}$$-cordial) or cordial (i.e. $${\mathcal Z}_ 2$$-cordial) graphs.
It has been conjectured by R. L. Graham and N. J. A. Sloane [SIAM J. Algebraic Discrete Methods 1, 382-404 (1980; Zbl 0499.05049)] that every tree $$T=(V,E)$$ is $${\mathcal Z}_{| E|}$$-cordial. In this paper it is shown that every tree is $${\mathcal Z}_ 3$$-, $${\mathcal Z}_ 4$$-, and $${\mathcal Z}_ 5$$-cordial. Several more results and conjectures are given.

##### MSC:
 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C05 Trees
##### Keywords:
cordial graphs; harmonic graphs; abelian group; labelling
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##### References:
 [1] Cahit, I., Cordial graphs: a weaker version of graceful and harmonious graphs, Ars. combin., 23, 201-207, (1987) · Zbl 0616.05056 [2] Chang, G.; Hsu, D.; Rogers, D., Additive variations on a graceful theme, Utilitas math., 32, 181-197, (1987) [3] S.M. Lee and A. Liu, On cordial graphs, to appear. · Zbl 0755.05086 [4] Grace, T., On sequential labelings of graphs, J. graph theory, 7, 195-201, (1987) · Zbl 0522.05063 [5] Graham, R.L.; Sloane, N.J.A., On additive bases and harmonious graphs, SIAM J. algebraic discrete methods, 1, 382-404, (1980) · Zbl 0499.05049
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