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 {H1, H2, ... , Hm}, denote G degrees Lambda i=1mHi, the generalized corona product of H and H1, H2, ... , Hm. In this paper, we prove that the total chromatic number of G degrees Lambda i=1mHi is the maximum of total chromatic number of G and maximum degree of G degrees Lambda i=1mHi is an element of plus one. As an immediate consequence, we prove that G degrees Lambda i=1mHi 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.)