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Antipodal Number of Cartesian Product of Complete Graphs with Cycles | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 14، دوره 10، شماره 1، خرداد 2025، صفحه 219-232 اصل مقاله (2.3 M) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28693.1674 | ||
| نویسندگان | ||
| Kush Kumar* ؛ Pratima Panigrahi | ||
| Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, West Bengal, India | ||
| چکیده | ||
| Let $G$ be a simple connected graph with diameter $d$, and $k\in [1,d]$ be an integer. A radio $k$-coloring of graph $G$ is a mapping $g:V(G)\rightarrow \{0\}\cup \mathbb{N}$ satisfying $\lvert g(u)-g(v)\rvert\geq 1+k-d(u,v)$ for any pair of distinct vertices $u$ and $v$ of the graph $G$, where $d(u,v)$ denotes distance between vertices $u$ and $v$ in $G$. The number ${\text{max}} \{g(u):u\in V(G)\}$ is known as the span of $g$ and is denoted by $rc_k(g)$. The radio $k$-chromatic number of graph $G$, denoted by $rc_k(G)$, is defined as $\text{min} \{rc_k(g) : g \text{ is a radio $k$-coloring of $G$}\}$. For $k=d-1$, the radio $k$-coloring of graph $G$ is called an antipodal coloring. So $rc_{d-1}(G)$ is called the antipodal number of $G$ and is denoted by $ac(G)$. Here, we study antipodal coloring of the Cartesian product of the complete graph $K_r$ and cycle $C_s$, $K_r\square C_s$, for $r\geq 4$ and $s\geq 3$. We determine the antipodal number of $K_r\square C_s$, for even $r\geq 4$ with $s\equiv 1(mod\,4)$; and for any $r\geq 4$ with $s=4t+2$, $t$ odd. Also, for the remaining values of $r$ and $s$, we give lower and upper bounds for $ac(K_r\square C_s)$. | ||
| کلیدواژه‌ها | ||
| radio $k$-coloring؛ antipodal coloring؛ antipodal number؛ Cartesian Product | ||
| مراجع | ||
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