Total Coloring of the Generalized Corona Product of Graphs

A long standing Total Coloring Conjecture (TCC) asserts that every graph is total colorable using its maximum degree plus two colors. A graph is type-1 (or type-2) if it has a total coloring using maximum degree plus one (or maximum degree plus two) colors. For a graph 𝐺 with 𝑚 vertices and for a family of graphs {𝐻 1 , 𝐻 2 , … , 𝐻 𝑚 }, denote , the generalized corona product of 𝐺 and 𝐻 1 , 𝐻 2 , … , 𝐻 𝑚 . In this paper, we prove that the total chromatic number of is the maximum of total chromatic number of 𝐺 and maximum degree of plus one. As an immediate consequence, we prove that is type-1 when 𝐺 satisfies TCC and also the corona product of 𝐺 and 𝐻 is type-1 if 𝐺 satisfies TCC. This generalizes some results in (R. Vignesh. et. al. in Discrete Mathematics, Algorithms and Applications, 11(1): 2019) and all the results in (Mohan et. al. in Australian Journal of Combinatorics, 68(1): 15-22, 2017.)