On Odd-Graceful Coloring of Graphs

Document Type : Original paper


1 Department of Mathematics, Universitas Pendidikan Ganesha, Singaraja-Bali, Indonesia

2 College of Engineering, Science and Environment, University of Newcastle, Australia

3 Universiti Malaysia Terengganu(UMT), Malaysia


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.


Main Subjects

[1] R. Alfarisi, R.M. Prihandini, R. Adawiyah, E.R. Albirri, and I.H. Agustin, Graceful chromatic number of unicyclic graphs, Journal of Physics: Conference Series, vol. 1306, IOP Publishing, 2019, pp. Article ID: 012039.
[2] A.D. Byers, Graceful Colorings and Connection in Graphs, Ph.D. thesis, Kalamazoo, Michigan, USA, 2018.
[3] S. English and P. Zhang, On graceful colorings of trees, Math. Bohem. 142 (2017), no. 1, 57–73.
[4] R.B. Gnanajothi, Topics in Graph Theory, Ph.D. thesis, Madurai, India, 1991.
[5] A.I. Kristiana, A. Aji, E. Wihardjo, and D. Setiawan, on graceful chromatic number of vertex amalgamation of tree graph family, CAUCHY: Jurnal Matematika Murni dan Aplikasi 7 (2022), no. 3, 432–444.
[6] R. Mincu, C. Obreja, and A. Popa, The graceful chromatic number for some particular classes of graphs, 2019 21st International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2019, pp. 109–115.
[7] A. Pasotti, On d-graceful graphs, Ars Combin. 111 (2013), 207–223.
[8] J. Su, H. Sun, and B. Yao, Odd-graceful total colorings for constructing graphic lattice, Mathematics 10 (2022), no. 1, Article ID: 109.
[9] H. Wang, J. Xu, and B. Yao, Twin odd-graceful trees towards information security, Procedia Computer Science 107 (2017), 15–20.