The results of section 4 on the independence number can be applied to study the chromatic number of *θ*-non-commuting graphs. Since every maximal independent set in $\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$ is a left coset to an abelian group, the evaluation of the chromatic number of
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$ relies on the theory of covering groups by left cosets of their proper subgroups. In this regard, the following result of Tomkinson plays an important role.

#### Theorem 5.1

(Tomkinson [7]). *Let* *G* *be covered by some cosets* *g*_{i}*H*_{i} *for* *i* = 1,…,*n*. *If the cover is irredundant*, *then*
$\begin{array}{}[G:\bigcap _{i=1}^{n}{H}_{i}]\le n!.\end{array}$

Tomkinson’s theorem has the following immediate result connecting the chromatic number of
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$ to the number of fixed points of *θ*.

#### Corollary 5.2

*For any group G*, *we have*
$$\begin{array}{}[G:\text{Fix}(\theta )]\le \chi ({\mathit{\Gamma}}_{G}^{\theta})!\end{array}$$

*and the equality holds only if* Fix(*θ*) ⊆ *Z*(*G*).

#### Proof

Let *G* = *I*_{1}∪⋯∪ *I*_{χ} be the union of independent sets *I*_{1},…,*I*_{χ} in which *I*_{i} ⊆ *g*_{i}A_{i} and *A*_{i} is an abelian subgroup of Fix(*I*_{gi}θ), for *i* = 1,…,*χ* = *χ*(
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$). If 1 ∈ *I*_{k}, then *g*_{k} ∈ *A*_{k}, which implies that *A*_{k} ⊆ Fix(*θ*). Thus
$$\begin{array}{}[G:\text{Fix}(\theta )]\le [G:{A}_{k}]\le [G:\bigcap _{i=1}^{n}{A}_{i}]\le \chi !.\end{array}$$

Now assume the equality holds. Then Fix(*θ*) = *A*_{k} ⊆ *A*_{i}, for all *i* = 1,…,*χ*. Since *A*_{i} are abelian, Fix(*θ*) commutes with all elements of *A*_{1},…,*A*_{χ}. On the other hand, as *A*_{k} ⊆ *A*_{i}, we have *a* = *θ*(*a*) =
$\begin{array}{}{a}^{{g}_{i}^{-1}}\end{array}$, for all *a* ∈ *A*_{k} and *i* = 1,…,*χ*, which implies that Fix(*θ*) commutes with *g*_{1},…,*g*_{χ} as well. Therefore Fix(*θ*) ⊆ *Z*(*G*), as required. □

In the sequel, we shall characterize those graphs having small chromatic numbers.

#### Theorem 5.3

*Let* *G* *be a finite group*. *Then* *χ*(
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$) = 2 *if and only if* *G* *is abelian and* [*G*:Fix(*θ*)] = 2.

#### Proof

Clearly, *α*(
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$) ≥ |*G*|/2. On the other hand, by Corollary 4.6, *α*(
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$) ≤ |*G*|/2, from which it follows that *α*(
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$) = |*G*|/2. Hence *G* = *I*_{1}∪ *I*_{2}, where (*I*_{1},*I*_{2}) is a bipartition of
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$ satisfying |*I*_{1}| = |*I*_{2}| = |*G*|/2. Assume 1 ∈ *I*_{1}. Then, by Theorem 4.5, *I*_{1} = *g*_{1}*A*_{1} for some *g*_{1} ∈ *G* in which *A*_{1} = Fix(*I*_{g1}*θ*) is an abelian subgroup of *G*. Since *g*_{1} ∈ *A*_{1}, it follows that *A*_{1} = Fix(*θ*). Clearly, *A*_{2} = *A*_{1} and *g*_{2} ∈ *G* ∖ *A*_{1}. Now
$\begin{array}{}\theta (a)={a}^{{g}_{1}^{-1}}={a}^{{g}_{2}^{-1}},\end{array}$ for all *a* ∈ *A*_{1}, from which it follows that *g*_{2} commutes with *A*_{1}. Thus *G* is abelian. The converse is obvious by Corollary 4.3. □

#### Theorem 5.4

*Let* *G* *be a finite group*. *Then* *χ*(
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$) = 3 *if and only if either* *G* *is abelian and* [*G*:Fix(*θ*)] = 3 *or* *G* *is non-abelian and one of the following holds*:

*G*/*Z*(*G*)≅ *C*_{2}× *C*_{2} *and* *θ* *is an inner automorphism*;

[*G*:Fix(*θ*)] = [Fix(*θ*):*Z*(*G*)] = 2; *or*

*G* *has a characteristic subgroup* *A* *such that* [*G*:*A*] = [*A*:Fix(*θ*)] = 2, Fix(*θ*) = *Z*(*G*) *and there exist elements* *x* ∈ *A* ∖ *Z*(*G*) *and* *y* ∈ *G* ∖ *A such that* *θ*(*x*) = *x*^{y}.

#### Proof

If *G* is abelian, then we are done by Corollary 4.3. Hence, we assume that *G* is non-abelian. Let *G* = *I*_{1}∪ *I*_{2}∪ *I*_{3} be a tripartition of *G* in which |*I*_{1}| ≥ |*I*_{2}| ≥ |*I*_{3}| and *I*_{i} ⊆ *g*_{i}A_{i} (*i* = 1,2,3) for some elements *g*_{i} ∈ *G* and abelian subgroups *A*_{i} of Fix(*I*_{gi}θ). Without loss of generality, we may assume that *I*_{1} = *g*_{1}*A*_{1}. From Theorem 5.1, we know that [*G*:*A*_{1}∩ *A*_{2}∩ *A*_{3}] ≤ 3! = 6. We distinguish two cases:

Case 1. *G* = 〈*A*_{1},*A*_{2},*A*_{3}〉. Then *A*_{1} ∩ *A*_{2} ∩ *A*_{3} ⊆ *Z*(*G*) and we must have *G*/*Z*(*G*)≅ *C*_{2}× *C*_{2} or *S*_{3}. Hence *A*_{1} ∩ *A*_{2} ∩ *A*_{3} = *Z*(*G*). One can verify that 2 = [*G*:*A*_{1}] ≥ [*G*:*A*_{2}] ≥ 3 and *A*_{1} ≠ *A*_{2}. Since *G* ≠ *A*_{1} ∪ *A*_{2} and every element of *G*/*Z*(*G*) has order 1, 2 or 3, one can always find an element *g* ∈ *G* ∖ *A*_{1} ∪ *A*_{2} such that *gA*_{i} = *g*_{i}A_{i}, for *i* = 1,2. Thus, for *a* ∈ *A*_{i} we have
$\begin{array}{}\theta (a)={a}^{{g}_{i}^{-1}}={a}^{{g}^{-1}}\end{array}$ which implies that *θ* acts by conjugation via *g*^{−1} on 〈*A*_{1},*A*_{2}〉 = *G*. Hence *θ* = *I*_{g−1} is an inner automorphism. If *G*/*Z*(*G*)≅ *S*_{3}, then
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$≅*Γ*_{G} has a subgraph isomorphic to *Γ*_{S3}≅ *K*_{5} ∖ *K*_{2} with chromatic number 4, a contradiction. Therefore *G*/*Z*(*G*)≅ *C*_{2}× *C*_{2}, which gives us part (1).

Case 2. *G* ≠〈*A*_{1},*A*_{2},*A*_{3}〉. Clearly, *A*_{2},*A*_{3} ⊆ *A*_{1}. Then *A*_{2},*A*_{3} ⊂ *A*_{1} otherwise *A*_{1} = *A*_{2} and hence *θ*(*a*) = *a*^{g1} = *a*^{g2} for all *a* ∈ *A*_{1}. Since *g*_{1}*A*_{1} ≠ *g*_{2}*A*_{2}, it follows that *g*_{1}
$\begin{array}{}{g}_{2}^{-1}\end{array}$ ∈ *G* ∖ *A*_{1} commutes with *A*_{1} so that *G* = *A*_{1}〈*g*_{1}
$\begin{array}{}{g}_{2}^{-1}\end{array}$〉 is abelian, a contradiction. Hence *A*_{2},*A*_{3} ⊂ *A*_{1}, from which together with Tomkinson’s result we must have [*G*:*A*_{1}] = 2, [*A*_{1}:*A*_{2}] = 2 and *A*_{2} = *A*_{3}. Clearly, *g*_{2}*A*_{2} ∪ *g*_{3}*A*_{3} =
$\begin{array}{}{g}_{1}^{{}^{\prime}}\end{array}$*A*_{1} where *G* = *g*_{1}*A*_{1} ∪
$\begin{array}{}{g}_{1}^{{}^{\prime}}\end{array}$*A*_{1}. If *g*_{1} ∈ *A*_{1}, then *A*_{1} = Fix(*θ*). As *A*_{2} ⊆ *A*_{1} we have *a* = *θ*(*a*) =
$\begin{array}{}{a}^{{g}_{2}^{-1}}\end{array}$ for all *a* ∈ *A*_{2}, which implies that *A*_{2} = *Z*(*G*). Hence we obtain part (2). Next assume that *g*_{1} ∉ *A*_{1}. Then
$\begin{array}{}{g}_{1}^{{}^{\prime}}\end{array}$ ∈ *A*_{1} and consequently *g*_{2},*g*_{3} ∈
$\begin{array}{}{g}_{1}^{{}^{\prime}}\end{array}$*A*_{1} = *A*_{1}. This implies that *A*_{2} ⊆ Fix(*θ*). As *A*_{2} ⊆ *A*_{1}, we have *a* = *θ*(*a*) =
$\begin{array}{}{a}^{{g}_{1}^{-1}}\end{array}$ for all *a* ∈ *A*_{2} showing that *A*_{2} = *Z*(*G*). Assuming *g*_{2} ∈ *A*_{1} ∖ *A*_{2}, we obtain *θ*(*g*_{2}) = *g*_{2}*z* for some *z* ∈ *A*_{2}. Furthermore,
$\begin{array}{}{g}_{2}z=\theta ({g}_{2})={g}_{2}^{{g}_{1}^{-1}}={g}_{2}^{{g}_{1}}\end{array}$ as *g*_{2} ∈ *A*_{1}. Thus [*g*_{2},*g*_{1}] = *z* and this yields part (3).

The converse is straightforward. □

We conclude this section with a characterization of complete multipartite-ness of the graphs
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$.

#### Theorem 5.5

*The graph*
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$ *is a complete multipartite graph if and only if* Fix(*I*_{g}θ) *is abelian for all g* ∈ *G*.

#### Proof

First assume that
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$ is a complete multipartite graph. From Theorem 4.5, it follows that all maximal abelian subgroups of Fix(*I*_{g}θ) are disjoint so that Fix(*I*_{g}θ) is abelian for all *g* ∈ *G*. Now assume that Fix(*I*_{g}θ) is abelian for all *g* ∈ *G*. Let *x* ∈ *G*. By assumption, *x*Fix(*I*_{x}θ) is an independent subset of
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$. On the other hand, for *y*∉ *x*Fix(*I*_{x}θ), we have *x*^{−1}*y*∉Fix(*I*_{x}θ) so that *θ*(*x*^{−1}*y*) ≠(*x*^{−1}*y*)^{x−1}. Hence *y* is adjacent to *x*. Since, by Theorem 4.5, the sets *g*Fix(*I*_{g}θ) are maximal independent subsets of
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$, it follows the sets *g*Fix(*I*_{g}θ) partition *G* and hence
$\begin{array}{}{\mathit{\Gamma}}_{G}^{\theta}\end{array}$ is a complete multipartite graph, as required. □

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