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Region indices of edge-friendly labeling of a plane graph. (English)
Util. Math. 99, 187-213 (2016).
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 $f: E\to{\bbfZ}_2$ induces a region labeling $f^ *: R\to{\bbfZ}_2$ defined as: $f *(r)= \sum_{e\in E_r-E’} f(e)\pmod 2$. For each, $i\in{\bbfZ}_2$ define $E_f(i)= |f^{-1}(i)$ and $R_f(i)= |f^{*-1}(i)|$. We call $f$ as edge-friendly if, $|E_f(1)- E_f(0)|\le 1$. 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 $P_m\times P_n$, $m\ge 2$, $n\ge 3$.