A note on obtaining bipartite radio graceful graphs of arbitrarily large radio numbers with radio graceful complements

Abstract

Motivated by the frequency assignment problem (FAP), a radio coloring of a graph $G$ is an assignment $f$ of non-negative integers to the vertices of $G$ satisfying the condition $|f\left(u\right)-f\left(v\right)|+d(u,v)\ge \text{diameter of}G+1$, where $d(u,v)$ is the distance between any two vertices $u$ and $v$ of the graph $G$. The span of a radio coloring of $G$ is the difference of the maximum and minimum non-negative integers used as colors. The minimum span of a radio coloring of $G$ is referred as the radio number of $G$. Any radio coloring with the minimum span is referred as an optimal radio coloring of $G$. If an optimal radio coloring of $G=(V,E)$ is a bijection from $V$ to $\{0,1,\dots ,\left|V\right|-1\}$, then the graph is referred as radio graceful. In this article, using a recursive construction, we have shown that for each positive integer $n\ge 9$, there exists a bipartite graph $G_n$ with $n+1$ vertices such that both $G_n$ and its complement $G_n^c$ are radio graceful graphs. In the process, we show that each such $G_n$ and $G_n^c$ contain a Hamiltonian path.

Note that our construction obtains radio graceful graphs of arbitrarily large radio numbers without going for big cliques. This has an interesting similarity with the motivation behind the Mycielski’s construction which ensures the existence of an infinite family of triangle- free graphs with arbitrarily large chromatic numbers. However, the growth of the vertex size in our recursive construction is linear unlike the Mycielski’s construction which yields exponential growth of the vertex size.