Fully Cordial Trees

Abstract

For a graph $G=(V,E)$ and a coloring $f:V(G)\to {\mathbb{Z}}_2$, let $v_f(i)=|f^{-1}(i)|$. $f$ is said to be friendly if $|v_f(1)–v_f(0)|\le 1$. The coloring $f:V(G)\to {\mathbb{Z}}_2$ induces an edge labeling $f_{+}:E(G)\to {\mathbb{Z}}_2$ defined by $f_{+}(xy)=|f(x)–f(y)|$, for all $xy\in E(G)$. Let $e_f(i)=|f_{+}^{-1}(i)|$. The friendly index set of the graph $G$, denoted by $FI(G)$, is defined by
$$FI(G)=\{|e_f(1)–e_f(0)|:f\text{ is a friendly vertex labeling of }G\}.$$

In this paper, we determine the friendly index set of certain classes of trees and introduce a few classes of fully cordial trees.