Geodesic
Equitable Total Coloring of Corona of Cubic Graphs
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 1147-1163

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The minimum number of total independent partition sets of V∪E of a graph G = (V, E) is called the total chromatic number of G, denoted by χ'′(G). If the difference between cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of V∪E is called the equitable total chromatic number, and is denoted by χ'′=(G). In this paper we consider equitable total coloring of coronas of cubic graphs, G◦H. It turns out that independently on the values of equitable total chromatic number of factors G and H, equitable total chromatic number of corona G◦H is equal to Δ(G◦H)+1. Thereby, we confirm Total Coloring Conjecture (TCC), posed by Behzad in 1964, and Equitable Total Coloring Conjecture (ETCC), posed by Wang in 2002, for coronas of cubic graphs. As a direct consequence we get that all coronas of cubic graphs are of Type 1.
Keywords: equitable coloring, total coloring, equitable total coloring, cubic graphs 
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Furmańczyk, Hanna; Zuazua, Rita. Equitable Total Coloring of Corona of Cubic Graphs. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 1147-1163. http://geodesic.mathdoc.fr/item/DMGT_2021_41_4_a18/