# Total Coloring Conjecture for Certain Classes of Graphs

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 2. Deleted Lexicographic Product

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

**Theorem**

**6.**

**Proof.**

**Note:**The above theorem is also holds good if we replace the path with any even cycles. Consider the two graphs G and H with m and n vertices respectively. If ${D}_{lex}(G,\overline{{K}_{n}})$ has a total coloring with $\Delta \left(G\right)+1$ colors such that the vertices of each $\overline{{K}_{n}}$ copy are colored pairwise distinctly and ${D}_{lex}(G,H)$ is total colorable. This can be proved very easily. ${D}_{lex}(G,\overline{{K}_{n}})={D}_{lex}(G,H)\backslash mH$, where $mH$ denotes the edges in the m copies of H. Color all the edges in m copies of H with $\Delta \left(H\right)+1$ colors and color all the elements of ${D}_{lex}(G,\overline{{K}_{n}})$ with $(n-1)\Delta \left(G\right)+1$ colors. This will give a total coloring of ${D}_{lex}(G,H)$.

## 3. Line Graphs and Double Graphs

#### Line Graphs

**Theorem**

**7.**

**Proof.**

**Conjecture**: For any complete graph ${K}_{n}$, ${\chi}^{\u2033}\left(L\left({K}_{n}\right)\right)=2n-3$.

**Theorem**

**8.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Vignesh, R.; Geetha, J.; Somasundaram, K. Total Coloring Conjecture for Certain Classes of Graphs. *Algorithms* **2018**, *11*, 161.
https://doi.org/10.3390/a11100161

**AMA Style**

Vignesh R, Geetha J, Somasundaram K. Total Coloring Conjecture for Certain Classes of Graphs. *Algorithms*. 2018; 11(10):161.
https://doi.org/10.3390/a11100161

**Chicago/Turabian Style**

Vignesh, R., J. Geetha, and K. Somasundaram. 2018. "Total Coloring Conjecture for Certain Classes of Graphs" *Algorithms* 11, no. 10: 161.
https://doi.org/10.3390/a11100161