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Region indices of edge-friendly labeling of a plane graph. (English) Zbl 1345.05090

Summary: Let \(G= (V,E)\) be a plane graph. If \(E'\) denotes the set of pendant edges and bridges, \(R\) the set of all regions in the graph \(G\) and \(E_r= \{e : e\) is an edge in region \(r\}\) then an edge labeling \(f: E\to\mathbb{Z}_2\) induces a region labeling \(f^ *: R\to\mathbb{Z}_2\) defined as: \(f *(r)= \sum_{e\in E_r-E'} f(e)\pmod 2\). For each, \(i\in\mathbb{Z}_2\) define \(E_f(i)= |f^{-1}(i)\) and \(R_f(i)= |f^{*-1}(i)|\). We call \(f\) as edge-friendly if, \(|E_f(1)- E_f(0)|\leq 1\). The full region index set of edge-friendly labeling of \(G\), \(FRIEFL(G)\) is defined as \(\{R_f(1)- R_f(0): f\) is edge-friendly labeling}.
In this paper, we study the full region index set of edge-friendly labeling of cycle \(C_n\), wheel \(W_n\), fans \(F_m\), \(F_{2,m}\), grid graph \(P_m\times P_n\), \(m\geq 2\), \(n\geq 3\).

MSC:

05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
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