On odd-graceful coloring of graphs

Document Type : Original paper

Authors

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

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

3 Special Interest Group on Modeling and Data Analytics (SIGMDA) Universiti Malaysia Terengganu, Malaysia

Abstract

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→{0,1,2,...,|E|} such that {|f(u)−f(v)|:uv∈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→{1,2,…,k}, for some positive integer k, which induces edge coloring |c(u)−c(v)|, uv∈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 Xog(G)=k. Otherwise, if G does not admit odd-graceful coloring, we will denote its odd-graceful chromatic number as Xog(G)=∞. In this paper, we derived some facts of odd-graceful coloring and determined odd-graceful chromatic numbers of some basic graphs.

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References

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Communications in Combinatorics and Optimization
Volume 10, Issue 2
June 2025
Pages 335-354

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