Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin.
Publication date
01/01/2023
Field of study
A graph G(V,E) is a system consisting of a finite non empty set of vertices V(G) and a set of edges E(G). A (proper) vertex colouring of G is a function f:V(G)→{1,2,…,k}, for some positive integer k such that f(u)=f(v) for every edge uv∈E(G). Moreover, if ∣f(u)−f(v)∣=∣f(v)−f(w)∣ for every adjacent edges uv,vw∈E(G), then the function f is called graceful colouring for G. The minimum number k such that f is a graceful colouring for G is called the graceful chromatic number of G. The purpose of this research is to determine graceful chromatic number of Cartesian product graphs Cm×Pn for integers m≥3 and n≥2, and Cm×Cn for integers m,n≥3. Here, Cm and Pm are cycle and path with m vertices, respectively. We found some exact values and bounds for graceful chromatic number of these mentioned Cartesian product graphs