@article{(Open Science Index):https://publications.waset.org/pdf/10008947, title = {Total Chromatic Number of Δ-Claw-Free 3-Degenerated Graphs}, author = {Wongsakorn Charoenpanitseri}, country = {}, institution = {}, abstract = {The total chromatic number χ"(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no incident or adjacent pair of elements receive the same color Let G be a graph with maximum degree Δ(G). Considering a total coloring of G and focusing on a vertex with maximum degree. A vertex with maximum degree needs a color and all Δ(G) edges incident to this vertex need more Δ(G) + 1 distinct colors. To color all vertices and all edges of G, it requires at least Δ(G) + 1 colors. That is, χ"(G) is at least Δ(G) + 1. However, no one can find a graph G with the total chromatic number which is greater than Δ(G) + 2. The Total Coloring Conjecture states that for every graph G, χ"(G) is at most Δ(G) + 2. In this paper, we prove that the Total Coloring Conjectur for a Δ-claw-free 3-degenerated graph. That is, we prove that the total chromatic number of every Δ-claw-free 3-degenerated graph is at most Δ(G) + 2.}, journal = {International Journal of Mathematical and Computational Sciences}, volume = {12}, number = {4}, year = {2018}, pages = {75 - 78}, ee = {https://publications.waset.org/pdf/10008947}, url = {https://publications.waset.org/vol/136}, bibsource = {https://publications.waset.org/}, issn = {eISSN: 1307-6892}, publisher = {World Academy of Science, Engineering and Technology}, index = {Open Science Index 136, 2018}, }