On Legendre Cordial Labeling of Complete Graphs

Let $p$ be an odd prime. For a simple connected graph $G$ of order $n$, a bijective function $f:V(G)\to \{1,2,\dots ,n\}$ is said to be a Legendre cordial labeling modulo $p$ if the induced function $f_p^{\ast }:E(G)\to \{0,1\}$, defined by $f_p^{\ast }(uv)=0$ whenever $([f(u)+f(v)]\mathrm{/}p)=-1$ or $f(u)+f(v)\equiv 0(\text{mod }p)$ and $f_p^{\ast }(uv)=1$ whenever $([f(u)+f(v)]\mathrm{/}p)=1$, satisfies the condition $\mathrm{\mid }{e}_{f_p^{\ast }}(0)-e_{f_p^{\ast }}(1)\mathrm{\mid }\le 1$ where $e_{f_p^{\ast }}(i)$ is the number of edges with label $i$ ($i=0,1$). This paper explores the characterization of the Legendre cordial labeling modulo $p$ of the complete graph $K_n$ using the concept of Legendre graph.