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Harmonious order of graphs. (English) Zbl 1188.05138
Summary: We consider the following generalization of the concept of harmonious graphs. Given a graph $$G=(V,E)$$ and a positive integer $$t\geq|E|$$, a function $$\widetilde{h}:V(G)\to\mathbb Z_t$$ is called a $$t$$-harmonious labeling of $$G$$ if $$\widetilde{h}$$ is injective for $$t\geq|V|$$ or surjective for $$t<|V|$$, and $$\widetilde{h}(v)+ \widetilde{h}(w)\neq \widetilde{h}(x)+ \widetilde{h}(y)$$ for all distinct edges $$vw,xy\in E(G)$$. Then the smallest possible $$t$$ such that $$G$$ has a $$t$$-harmonious labeling is named the harmonious order of $$G$$. We determine the harmonious order of some non-harmonious graphs, such as complete graphs $$K_n$$ $$(n\geq5)$$, complete bipartite graphs $$K_{m,n}$$ $$(m,n>1)$$, even cycles $$C_n$$, some powers of paths $$P_n^k$$, disjoint unions of triangles $$nK_3$$ ($$n$$ even). We also present some general results concerning harmonious order of the Cartesian product of two given graphs or harmonious order of the disjoint union of copies of a given graph. Furthermore, we establish an upper bound for harmonious order of trees.

##### MSC:
 05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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##### References:
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