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On edge-balanced index sets of some complete $$k$$-partite graphs. (English) Zbl 1211.05149
Summary: Let $$G= (V,E)$$ be an undirected graph. A binary labeling $$f: E\to Z_2= \{0,1\}$$ is edge-friendly if $$|e_f(0)- e_f(1)|\leq 1$$, where $$e_f(i)$$ is the number of $$i$$-edges labeled with $$i$$, $$i\in Z_2$$. Given any edge-friendly labeling $$f$$, $$f$$ induces a partial vertex labeling $$f^+$$ on $$V$$ defined by $$f^+(v)= 0,\,1$$, or undefined, iff the number of $$0$$-edges incident with $$x$$ is greater than, less than, or equal to, the number of $$1$$-edges incident with $$x$$. Let $$\Sigma$$ be the set of all edge-friendly labelings of $$G$$. For $$f\in E$$, the edge-balanced index of $$f$$ is defined as $$\delta(f)=|v_f(0)- v_f(1)|$$, where $$v_f(i)$$ is the number of $$i$$-vertices labeled with $$i$$, $$i\in Z_2$$. The edge-balance index set of $$G$$ is defined as $$\text{EBI}(G)= \{\delta(f): f\in\Sigma\}$$. If $$0\in \text{EBI}(G)$$ or $$1\in\text{EBI}(G)$$, then $$G$$ is an edge-balanced graph.
In this paper we determine the edge-balance index set of some complete $$k$$-partite graphs.

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