Abstract
A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by χ ″ ( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, Δ ( G ) + 1 ≤ χ ″ ( G ) ≤ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.
Highlights
All the graphs in this paper are finite, simple and connected
The edge chromatic number of a graph G, denoted by χ0 ( G ), is the smallest number of colors needed to color the edges of G so that no two adjacent edges share the same color
McDiarmind and Sánchez-Arroyo [4] proved that determining the total chromatic number is NP-hard even for μ-regular bipartite graphs, for each fixed μ ≥ 3
Summary
All the graphs in this paper are finite, simple and connected. The edge chromatic number of a graph G, denoted by χ0 ( G ) , is the smallest number of colors needed to color the edges of G so that no two adjacent edges share the same color. McDiarmind and Sánchez-Arroyo [4] proved that determining the total chromatic number is NP-hard even for μ-regular bipartite graphs, for each fixed μ ≥ 3. The TCC was verified for graph products, such as Cartesian and Direct products, of certain classes of graphs. Seoud [7,8] proved that the Cartesian product graphs Pm Pn , m, n ≥ 2, except P2 P2 are of type I. Mohan et al [15] proved that certain classes of Corona product graphs are type-I. In [16], they proved the TCC for certain classes of product graphs. We prove the TCC for certain classes of deleted lexicographic product. We obtain results on the total chromatic number for line graphs, which is a subclass of claw-free graphs.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
