Results on graceful chromatic number for particular graphs

In graph theory, graph colorings are a major area of study. Graph colorings involve the constrained assignment of labels (colors) to vertices or edges. There are many types of colorings defined in the literature, the most common being the proper vertex coloring. The proper vertex $k$-coloring is defined as a vertex coloring from a set of $k$ colors such that no two adjacent vertices have the same color. In this paper, we focus on a variant of the proper vertex $k$-coloring problem, termed graceful coloring. A graceful $k$-coloring of an undirected connected graph $G$ is a proper vertex coloring using $k$ colors, that induces a proper edge coloring, where the color for an edge ($u, v$) is the absolute value of the difference between the colors assigned to vertices $u$ and $v$. The minimum $k$ for which a graph $G$ has a graceful $k$-coloring is termed the graceful chromatic number of the graph. In a previous work (Mincu, Obreja, Popa, SYNASC 2019) we find the graceful chromatic number for some well-known graphs and classes of graphs, such as diamond graph, Petersen graph, Moser spindle graph, Goldner-Harary graph, friendship graphs, fan graphs, and others. In this study, we continue the investigation and find the graceful chromatic number for other well-known individual graphs, like Dürer graph, Heawood graph, Möbius-Kantor graph, Nauru graph, Tietze's graph, Golomb graph and classes of graphs, like cactus, Gear, web graphs, etc. In a previous work (Mincu, Obreja, Popa, SYNASC 2019) we find the graceful chromatic number for some well-known graphs and classes of graphs, such as diamond graph, Petersen graph, Moser spindle graph, Goldner-Harary graph, friendship graphs, fan graphs, and others. In this study, we continue the investigation and find the graceful chromatic number for other well-known individual graphs, like Dürer graph, Heawood graph, Möbius-Kantor graph, Nauru graph, Tietze's graph, Golomb graph and classes of graphs, like cactus, Gear, web graphs, etc.