Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 9 2 2024 06 01 Total Chromatic Number for Certain Classes of Lexicographic Product Graphs 233 240 14478 10.22049/cco.2022.27736.1333 EN T.P. Sandhiya Department of Mathematics, Amrita School of Physical Sciences - Coimbatore, Amrita Vishwa Vidyapeetham, India J. Geetha Department of Mathematics, Amrita School of Physical Sciences - Coimbatore, Amrita Vishwa Vidyapeetham, India K. Somasundaram Department of Mathematics, Amrita School of Physical Sciences - Coimbatore, Amrita Vishwa Vidyapeetham, India Journal Article 2022 03 28 A total coloring of a graph $G$ is an assignment of colors to all the elements (vertices and edges) of the graph in such a way that no two adjacent or incident elements receive the same color. The total chromatic number of $G$, denoted by $\chi''(G)$, is the minimum number of colors which need for total coloring of $G$. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing which claims that, $\Delta(G)+1 \leq \chi''(G) \leq \Delta(G)+2 $, where $\Delta(G)$ is the maximum degree of $G$. The lower bound is sharp and the upper bound remains to be proved. In this paper, we prove the TCC for certain classes of lexicographic and deleted lexicographic products of graphs. Also, we obtained the lower bound for certain classes of these products. https://comb-opt.azaruniv.ac.ir/article_14478_6ae5822cff5fd512923619eaae0409a8.pdf