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T. Kavaskar Department of Mathematics, Central University of Tamil Nadu, Thiruvarur - 610 005, INDIA

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Sreelakshmi Sukumaran Department of Mathematics, Central University of Tamil Nadu, Thiruvarur - 610 005, INDIA

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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˜Λi=1mHi, the generalized corona product of 𝐺 and 𝐻1, 𝐻2, … , 𝐻𝑚. In this paper, we prove that the total chromatic number of G˜Λi=1mHi is the maximum of total chromatic number of 𝐺 and maximum degree of G˜Λi=1mHi plus one. As an immediate consequence, we prove that G˜Λi=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.)

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
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Size B5
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1966
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Founder Magyar Tudományos Akadémia  
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ISSN 0081-6906 (Print)
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