### Harmonious order of graphs

Andrzej Żak
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We consider the following generalization of the concept of harmonious graphs. Given a graph G = (V , E) and a positive integer t ≥ |E|, a functionh : V (G) → Z t is called a t-harmonious labeling of G ifh is injective for t ≥ |V | or surjective for t < |V |, and h(v) +h(w) =h(x) +h(y) for all distinct edges vw, xy ∈ 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
more &raquo; ... s complete graphs K n (n ≥ 5), complete bipartite graphs K m,n (m, n > 1), even cycles C n , some powers of paths P k n , 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.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2009.05.010">doi:10.1016/j.disc.2009.05.010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xum4y6xnibabfnn2o4tuvtjcta">fatcat:xum4y6xnibabfnn2o4tuvtjcta</a> </span>
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