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 with vertices and for a family of graphs {1, 2, ... , }, denote G ∘ ˜ Λ i = 1 m H i , the generalized corona product of and 1, 2, ... , . In this paper, we prove that the total chromatic number of G ∘ ˜ Λ i = 1 m H i is the maximum of total chromatic number of and maximum degree of G ∘ ˜ Λ i = 1 m H i ∈ plus one. As an immediate consequence, we prove that G ∘ ˜ Λ i = 1 m H i is type-1 when satisfies TCC and also the corona product of and is type-1 if 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.) [ABSTRACT FROM AUTHOR]