Edge-cordial labeling

Edge-cordial labeling --- рёберно-сердечная разметка. Yilmaz and Cahit introduced in 1997 edge-cordial labeling as a weaker version of edge graceful labeling. Let [math]\displaystyle{ f }[/math] be a binary edge labeling of graph [math]\displaystyle{ G = (V,E) }[/math]; that is, [math]\displaystyle{ f: \; E \rightarrow \{0,1\} }[/math]. Let the induced vertex labeling be given by [math]\displaystyle{ f^{+}(v) = \sum_{uv \in E}f(uv)\pmod{2} }[/math], where [math]\displaystyle{ v \in V }[/math]. The function [math]\displaystyle{ f }[/math] is called an edge-cordial labeling of [math]\displaystyle{ G }[/math] if the following two properties hold:

(1) [math]\displaystyle{ |e_{f}(0) - e_{f}(1)| \leq 1 }[/math], and

(2) [math]\displaystyle{ |v_{f}(0) - v_{f}(1)| \leq 1 }[/math],

where [math]\displaystyle{ e_{f}(0) }[/math] and [math]\displaystyle{ e_{f}(1) }[/math] denote the number of edges, and [math]\displaystyle{ v_{f}(0) }[/math] and [math]\displaystyle{ v_{f}(1) }[/math] denote the number of vertices labeled with 0 or 1, respectively. The graph [math]\displaystyle{ G }[/math] is called edge-cordial if it admits an edge-cordial labeling.