Extreme edge-friendly indices of complete bipartite graphs

Abstract

Let G=(V,E) be a simple graph. An edge labeling f:E to {0,1} induces a vertex labeling f^+:V to Z_2 defined by $f^+(v)equiv sumlimits_{uvin E} f(uv)pmod{2}$ for each $v in V$, where Z_2={0,1} is the additive group of order 2. For $iin{0,1}$, let e_f(i)=|f^{-1}(i)| and v_f(i)=|(f^+)^{-1}(i)|. A labeling f is called edge-friendly if $|e_f(1)-e_f(0)|le 1$. I_f(G)=v_f(1)-v_f(0) is called the edge-friendly index of G under an edge-friendly labeling f. Extreme values of edge-friendly index of complete bipartite graphs will be determined.

Keywords

• Edge-friendly index
• edge-friendly labeling
• complete bipartite graph