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On Odd-Graceful Coloring of Graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 27 آبان 1402 اصل مقاله (600.42 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28736.1692 | ||
| نویسندگان | ||
| I Nengah Suparta* 1؛ Yuqing Lin2؛ Roslan Hasni3؛ I Nyoman Budayana1 | ||
| 1Department of Mathematics, Universitas Pendidikan Ganesha, Singaraja-Bali, Indonesia | ||
| 2College of Engineering, Science and Environment, University of Newcastle, Australia | ||
| 3Universiti Malaysia Terengganu(UMT), Malaysia | ||
| چکیده | ||
| For a graph $G(V,E)$ which is undirected, simple, and finite, we denote by $|V|$ and $|E|$ the cardinality of the vertex set $V$ and the edge set $E$ of $G$, respectively. A \textit{graceful labeling} $f$ for the graph $G$ is an injective function ${f}:V\rightarrow \{0,1,2,..., |E|\}$ such that $\{|f(u)-f(v)|:uv\in E\}=\{1,2,...,|E|\}$. A graph that has a graceful-labeling is called \textit{graceful} graph. A vertex (resp. edge) coloring is an assignment of color (positive integer) to every vertex (resp. edge) of $G$ such that any two adjacent vertices (resp. edges) have different colors. A \textit{graceful coloring} of $G$ is a vertex coloring $c: V\rightarrow \{1,2,\ldots, k\},$ for some positive integer $k$, which induces edge coloring $|c(u)-c(v)|$, $uv\in E$. If $c$ also satisfies additional property that every induced edge color is odd, then the coloring $c$ is called an \textit{odd-graceful coloring} of $G$. If an odd-graceful coloring $c$ exists for $G$, then the smallest number $k$ which maintains $c$ as an odd-graceful coloring, is called \textit{odd-graceful chromatic number} for $G$. In the latter case we will denote the odd-graceful chromatic number of $G$ as $\mathcal{X}_{og}(G)=k$. Otherwise, if $G$ does not admit odd-graceful coloring, we will denote its odd-graceful chromatic number as $\mathcal{X}_{og}(G)=\infty$. In this paper, we derived some facts of odd-graceful coloring and determined odd-graceful chromatic numbers of some basic graphs. | ||
| کلیدواژه‌ها | ||
| Graceful graph؛ graceful coloring؛ odd-graceful coloring؛ odd-graceful chromatic number | ||
| مراجع | ||
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