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A STUDY ON $r$-HUED COLOURING OF PRISM GRAPH FAMILIES
Keywords:
r-hued chromatic number, n-crossed prism graph, line graph, antiprism graph.DOI:
https://doi.org/10.17654/0974165825033Abstract
Let $G$ be a simple, connected, undirected, and finite graph with vertex set $V(G)$ and edge set $E(G)$. For a positive integer $r$, a proper $k$-colouring of $G$ is called an $r$-hued colouring if, for each vertex $z \in V(G), \quad|f(N(z))| \geq \min \{r, d(z)\}$, where $N(z)$ denotes the neighbourhood of $z$ and $d(z)$ is the degree of $z$. The minimum $k$ that allows an $r$-hued colouring of $G$ with $k$ colours is referred to as the $r$-hued chromatic number of $G$, denoted by $\chi_r(G)$. In this paper, we determine the $r$-hued chromatic number for several graph classes: the $n$-crossed prism graph $R_n$, the line graph of $R_n$, denoted $L\left(R_n\right)$, the prism graph $C L_n$, and the antiprism graph $Q_n$.
Received: February 25, 2025
Revised: March 5, 2025
Accepted: April 8, 2025
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