DOI: 10.15672/hujms.1637391 ISSN: 2651-477X

Antipodal coloring number of graphs with specified triameter

Kush Kumar, Pratima Panigrahi
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Given a graph $\mathcal{G}$ on $\mathfrak{n}$ vertices and diameter $d$, a radio $k$-coloring $\mathfrak{h}$ is an assignment of non-negative integers to the vertices of $\mathcal{G}$ such that for every pair of vertices $\mathfrak{u}$ and $\mathfrak{v}$, the condition $|\mathfrak{h}(\mathfrak{u}) - \mathfrak{h}(\mathfrak{v})| \geq 1 + k - \mathfrak{d}(\mathfrak{u}, \mathfrak{v})$ holds, where $\mathfrak{d}(\mathfrak{u}, \mathfrak{v})$ is the distance between vertices $\mathfrak{u}$ and $\mathfrak{v}$. The span of a radio $k$-coloring $\mathfrak{h}$ is the largest integer assigned by $\mathfrak{h}$. The radio $k$-chromatic number of $\mathcal{G}$, denoted $rc_k(\mathcal{G})$, is the minimum span among all possible radio $k$-colorings of $\mathcal{G}$. For $k=d-1$, the radio $k$-coloring is known as an antipodal coloring, and the number $rc_{d-1}(\mathcal{G})$ is called the antipodal number of $\mathcal{G}$, also denoted by $ac(\mathcal{G})$. In this work, we derive the antipodal number of graphs having triameter less than or equal to $2d+1$, or equal to $3d$ allowing a particular vertex ordering with respect to distances. Then, applying these results and obtain the antipodal number for the Generalized Petersen graphs $GP(4\mathfrak{t}+2,2)$, $GP(4\mathfrak{t},2)$ for $\mathfrak{t}\geq3$ and Hamming graphs $H(3,3)$ and $H(4,3)$.

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