# Connectivity and Hamiltonicity of Canonical Colouring Graphs of Bipartite and Complete Multipartite Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Background, and a Preliminary Result

**Proposition**

**1.**

**Proof.**

## 3. Unions and Joins

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Proposition**

**2.**

**Corollary**

**4.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Corollary**

**7.**

**Proof.**

## 4. Bipartite Graphs

**Theorem**

**3.**

**Proof.**

- If $m\le \left|B\right|+1$ then all vertices coloured k in ${s}_{km}$ are in B so the set is independent.
- If $m>\left|B\right|+1$, this means that the only vertices in B that are still coloured k are coloured k under c, that is: $\left|B\right|\cap \left\{{v}_{j}\right|{s}_{km}\left({v}_{j}\right)=k\}=\left|B\right|\cap \left\{{v}_{j}\right|c\left({v}_{j}\right)=k\}$. No vertices in A are coloured k under ${c}^{\prime}$ so if ${v}_{j}\in A$ and ${s}_{km}\left({v}_{j}\right)=k$, then $m>j$ and ${s}_{km}\left({v}_{j}\right)=c\left({v}_{j}\right)=k$. Thus $\left\{{v}_{j}\right|{s}_{km}\left({v}_{j}\right)=k\}\subseteq \left\{{v}_{j}\right|c\left({v}_{j}\right)=k\}$ which is independent.

**Proposition**

**4.**

- 1.
- ${\mathrm{Can}}_{k}^{\pi}\left(G\right)$ has a cut vertex and hence has no Hamilton cycle;
- 2.
- if $r\ge 3$ then ${\mathrm{Can}}_{k}^{\pi}\left(G\right)$ has no Hamilton path.

**Proof.**

**Theorem**

**4.**

- (i)
- the colouring ${x}_{1}=11\dots 1$, and the colouring ${x}_{t}$ uses all k colours.
- (ii)
- For each $1<i<t$, the set of colours used by ${x}_{i}$ is identical to the set used by either ${x}_{i-1}$, ${x}_{i+1}$.

**Proof.**

**Theorem**

**5.**

**Proof.**

## 5. ${\mathrm{Can}}_{\mathit{k}}^{\mathit{\pi}}\left({\mathit{T}}_{\mathbf{2}\mathit{n},\mathit{n}}\right)$

**Theorem**

**6.**

- 1.
- ${\mathrm{Can}}_{n}^{\pi}\left({T}_{2n,n}\right)\cong {K}_{1}$ for any vertex ordering π.
- 2.
- If $k\ge 2n$, then ${\mathrm{Can}}_{k}^{\pi}\left({T}_{2n,n}\right)\cong {\mathrm{Can}}_{2n}^{\pi}\left({T}_{2n,n}\right)$ for any vertex ordering π.
- 3.
- If $n<k$ and the subgraph of ${T}_{2n,n}$ induced by the first n vertices in the vertex ordering π is not complete, then ${\mathrm{Can}}_{k}^{\pi}\left({T}_{2n,n}\right)$ is disconnected.
- 4.
- If $n<k$ and the subgraph of ${T}_{2n,n}$ induced by the first n vertices in the vertex ordering π is complete, then ${\mathrm{Can}}_{k}^{\pi}\left({T}_{2n,n}\right)$ is a tree. Further, if ${\mathrm{Can}}_{k}^{\pi}\left({T}_{2n,n}\right)$ and ${\mathrm{Can}}_{k}^{\varphi}\left({T}_{2n,n}\right)$ are both trees, then ${\mathrm{Can}}_{k}^{\pi}\left({T}_{2n,n}\right)\cong {\mathrm{Can}}_{k}^{\varphi}\left({T}_{2n,n}\right)$.
- 5.
- ${\mathrm{Can}}_{2n}^{\pi}\left({T}_{2n,n}\right)$ never has a Hamilton cycle and has a Hamilton path only when $n=2$, $k=2$.

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Three different vertex orderings of ${P}_{4}$ with associated $Ca{n}_{3}^{\pi}\left({P}_{4}\right)$. In each case the colourings are canonical with respect to the given vertex ordering from left to right.

**Figure 2.**${\mathcal{C}}_{3}\left({P}_{4}\right)$, the 3-Colouring Graph of ${P}_{4}$. The vertices are labeled by the colourings of the path.

**Figure 3.**${\mathcal{B}}_{3}\left({P}_{4}\right)$, the 3-Bell colouring graph of ${P}_{4}$. The vertices are labeled by the partition of the path $abcd$.

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**MDPI and ACS Style**

Haas, R.; MacGillivray, G.
Connectivity and Hamiltonicity of Canonical Colouring Graphs of Bipartite and Complete Multipartite Graphs. *Algorithms* **2018**, *11*, 40.
https://doi.org/10.3390/a11040040

**AMA Style**

Haas R, MacGillivray G.
Connectivity and Hamiltonicity of Canonical Colouring Graphs of Bipartite and Complete Multipartite Graphs. *Algorithms*. 2018; 11(4):40.
https://doi.org/10.3390/a11040040

**Chicago/Turabian Style**

Haas, Ruth, and Gary MacGillivray.
2018. "Connectivity and Hamiltonicity of Canonical Colouring Graphs of Bipartite and Complete Multipartite Graphs" *Algorithms* 11, no. 4: 40.
https://doi.org/10.3390/a11040040