Bounds for boxicity of circular clique graphs and zero-divisor graphs

Abstract

Let $box\left(G\right)$ be the boxicity of a graph $G$, $G[H_1,H_2,\dots ,H_n]$ be the $G$-join graph of $n$-pairwise disjoint graphs $H_1,H_2,\dots ,H_n$, $G_k^d$ be a circular clique graph (where $k\ge 2d$) and $\Gamma \left(R\right)$ be the zero-divisor graph of a commutative ring $R$ with unity. In this paper, we prove that $\chi \left(G_k^d\right)\ge box\left(G_k^d\right)$, for all $k$ and $d$ with $k\ge 2d$. This generalizes the results proved by Kamibeppu (2018). Also we obtain that $box\left(G[H_1,H_2,\dots ,H_n]\right)\le {\sum }_{i=1}^{n}box\left(H_i\right)$. As a consequence of this result, we obtain a bound for boxicity of ideal-based zero-divisor graph of a finite commutative ring with unity. In particular, if $R\ncong Z_2\times F_q$ is a finite commutative non-zero reduced ring with unity, where $F_q$ is a finite filed, then $\chi \left(\Gamma \left(R\right)\right)\le box\left(\Gamma \left(R\right)\right)\le 2^{\chi \left(\Gamma \left(R\right)\right)}-2$. where $\chi \left(\Gamma \left(R\right)\right)$ is the chromatic number of $\Gamma \left(R\right)$. Moreover, we show that if $N={\prod }_{i=1}^a{p}_i^{2n_i}{\prod }_{j=1}^b{q}_j^{2m_j+1}$ is a composite number, where $p_i$’s and $q_j$’s are distinct prime numbers, then $box\left(\Gamma \left(Z_N\right)\right)\le \left({\prod }_{i=1}^a(2n_i+1){\prod }_{j=1}^b(2m_j+2)\right)-\left({\prod }_{i=1}^a(n_i+1){\prod }_{j=1}^b(m_j+1)\right)-1$, where $Z_N$ is the ring of integers modulo $N$. Further, we prove that, $box\left(\Gamma \left(Z_N\right)\right)=1$ if and only if either $N=p^n$ for some prime number $p$ and some positive integer $n\ge 3$ or $N=2p$ for some odd prime number $p$.