**1. Introduction**

Cayley graphs on a group and a generating set have been an important class of graphs in the study of interconnection networks for parallel and distributed computing [1–6]. Some recent results about topological properties and routing problems on the networks based on Cayley graphs on the symmetric groups with the set of transpositions as the generating sets, including two special classes, the star graphs [5] and bubble-sort graphs [1], can be found in [6–9].

Throughout this paper, we consider finite, simple connected graph. Let Γ be a graph with vertex set *V*(Γ) and edge set *E*(Γ). A graph *H* is a subgraph of Γ if *V*(*H*) ⊆ *V*(Γ) and *E*(*H*) ⊆ *E*(Γ). The induced subgraph Γ[*C*] is the subgraph of Γ with vertex set *C* and edge set {*uv*|*u*, *v* ∈ *C*, *uv* ∈ *E*(Γ)}. Let *G* be a group, *S* a subset of *G* such that the identity element does not belong to *S* and *S* = *S*<sup>−</sup>1, where *S*−<sup>1</sup> = {*τ*−1|*τ* ∈ *S*}. The *Cayley graph* Γ, denoted by Γ = Cay(*G*, *S*), is the graph whose vertex set *V*(Γ) = *G* and *u*, *v* are adjacent if and only if *u*−1*v* ∈ *S*. It's known that Γ is connected if and only if *S* is a generating set of *G*. Furthermore, obviously, all Cayley graphs are vertex-transitive (see [10]).

We denote S*<sup>n</sup>* as the symmetric group on *n* letters (set of all permutations on {1, 2, ... , *n*}). Now let us restrict *S* to be a subset of transpositions on {1, 2, ... , *n*}. Clearly all Cayley graphs Cay(S*n*, *S*) are |*S*|-regular bipartite graphs. The *transposition generating graph* of *S*, denoted by *T*(*S*), is the graph with vertex set {1, 2, ... , *n*} and two vertices *s* and *t* are adjacent if and only if the transposition (*st*) is in *S*. If *T*(*S*) is a tree, it is called *transposition trees*.

An edge set *M* ⊆ *E*(Γ) is called a *matching* of Γ if no two of them share an end-vertex. Moreover, a matching of Γ is said to be *per f ect* if it covers all vertices of Γ. A connected graph Γ having at least 2*k* + 2 vertices is said to be *k-extendable*, introduced by Plummer [11], if each matching *M* of *k* edges is contained in a perfect matching of Γ. Any *k*-extendable graph is (*k* − 1)-extendable, but the converse is not true [11]. The *extendability number* of Γ, denoted by *ext*(Γ), is the maximum *k* such that Γ is *k*-extendable. Plummer [11,12] studied the relationship between *n*-extendability and other graph properties. For more research results related to matching extendability, one can refer to [13–17]. Yu et al. [18]

**Citation:** Feng, Y.; Xie, Y.; Liu, F.; Xu, S. The Extendability of Cayley Graphs Generated by Transposition Trees. *Mathematics* **2022**, *10*, 1575. https://doi.org/10.3390/ math10091575

Academic Editors: Irina Cristea and Hashem Bordbar

Received: 26 March 2022 Accepted: 3 May 2022 Published: 7 May 2022

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classified the 2-extendable Cayley graphs of finite abelian groups. Chen et al. [19] classified the 2-extendable Cayley graphs of dihedral groups. Recently, Gao et al. [20] characterize the 2-extendable quasi-abelian Cayley graphs. Their research is focused on 2-extendability of some Cayley graphs; in this paper, we focus on the general extendability, i.e., (*n* − 2) extendability of Cayley graphs generated by transposition trees.

We proceed as follows. In Section 2, we provide preliminaries and previous related results on Cayley graphs. In Section 3, we give our main results: show that all Cayley graphs generated by transposition trees are (*n* − 2)-extendable and then determine their extendability numbers are *n* − 2.

#### **2. Preliminaries**

In this section, we shall give some definitions and known results which will be used in this paper.

Denote by S*<sup>n</sup>* the group of all permutations on [*n*] = {1, 2, ... , *n*}. Obviously, |S*n*| = *n*!. For convenience, we use **x** = *x*1*x*<sup>2</sup> ... *xn* to denote the permutation ( 1 2 ... *<sup>n</sup> x*<sup>1</sup> *x*2... *xn* ) (see [21]); (*st*) to denote the permutation ( 1...*s*...*t*...*n* 1...*t*...*s*...*n*), which is called a *transposition*. Obviously, *x*<sup>1</sup> ... *xs* ... *xt* ... *xn*(*st*) = *x*<sup>1</sup> ... *xt* ... *xs* ... *xn*. The identity permutation 12 ... *n* is denoted by **1** . A permutation of S*<sup>n</sup>* is said to be *even* (resp. *odd*) if it can be written as a product of an even (resp. odd) number of transpositions. Let *S* be a subset of transpositions. Clearly, the Cayley graph Cay(S*n*, *S*) is a bipartite graph with one partite set containing the vertices corresponding to odd permutations and the other partite set containing the vertices corresponding to even permutations.

To better describe a transposition set *S* as the generating set, we use a simple way to depict *S* via a graph. The *transposition generating graph T*(*S*) is the graph with vertex set [*n*] and two vertices *s* and *t* are adjacent if and only if (*st*) ∈ *S*. If *T*(*S*) is a tree, it is called *transposition trees*, we denote by T*<sup>n</sup>* the set of Cayley graphs Cay(S*n*, *S*) generated by transposition trees. For any graph T*n*(*S*) = Cay(S*n*, *S*) ∈ T*n*, **x** = *x*1*x*<sup>2</sup> ... *xn* is adjacent to **y** = *y*1*y*<sup>2</sup> ... *yn* if and only if for (*st*) ∈ *S*, *xs* = *yt*, *xt* = *ys* and *xk* = *yk* for *k* = *s*, *t*, that is **y** = **x**(*st*). In this case, we say that the edge *e* = **xy** is an (*st*)-edge and denote *g*(*e*)=(*st*), which is the edge *e* corresponding to transposition. Let *Est* = {*e* ∈ *E*(T*n*(*S*))|*e* is an (*st*)-edge}. Obviously, for every transposition (*st*) ∈ *S*, *Est* is a perfect matching of T*n*(*S*). We have the following propositions about Cayley graphs generated by transpositions:

**Proposition 1** ([10], p. 52)**.** *Let* Γ = Cay(S*n*, *S*) *be a Cayley graph generated by transpositions. Then,* Γ *is connected if and only if T*(*S*) *is connected.*

**Proposition 2** ([22])**.** *Let S and S be two sets of transpositions on* [*n*]*. Then,* Cay(S*n*, *S*) *and* Cay(S*n*, *S* ) *are isomorphic if and only if T*(*S*) *and T*(*S* ) *are isomorphic.*

In all Cayley graphs T*n*, there are two classes which are most important, when *T*(*S*) is isomorphic to the star *<sup>K</sup>*1,*n*−<sup>1</sup> and the path *Pn*. If *<sup>T</sup>*(*S*) ∼= *<sup>K</sup>*1,*n*<sup>−</sup>1, Cay(S*n*, *<sup>S</sup>*) is called *ndimensional star graph* and denoted by *STn*. If *T*(*S*) ∼= *Pn*, Cay(S*n*, *S*) is called *n-dimensional bubble-sort graph* and denoted by *BSn*. The star graph and the bubble-sort graph are illustrated in Figures 1 and 2 for the case *n* = 4. Both *STn* and *BSn* are connected bipartite (*<sup>n</sup>* − 1)-regular graph of order *<sup>n</sup>*!. When *<sup>n</sup>* = 3, T3(*S*) ∼= *ST*<sup>3</sup> ∼= *BS*<sup>3</sup> ∼= *<sup>C</sup>*6; *<sup>n</sup>* = 4, up to isomorphism, there are exactly two different graphs *ST*<sup>4</sup> and *BS*<sup>4</sup> (see [23]).

**Figure 1.** The star graph *ST*<sup>4</sup> = Cay(S4, {(12),(13),(14)}).

**Figure 2.** The Bubble-sort graph *BS*<sup>4</sup> = Cay(S4, {(12),(23),(34)}).

Let **x** = *x*1*x*<sup>2</sup> ... *xn* be a vertex of T*n*(*S*). We say that *xi* is the *i-th coordinate* of **x**, denoted by (**x**)*i*. It is easy to see that the Cayley graph T*n*(*S*) has the following proposition:

**Proposition 3** ([23,24])**.** *Let <sup>T</sup>*(*S*) *be a transposition tree of order n, <sup>j</sup> one of its leaf and* <sup>T</sup> {*i*} *<sup>n</sup>* (*S*) (1 ≤ *i* ≤ *n*) *the subgraph of* T*n*(*S*) *induced by those vertices* **x** *with* (**x**)*<sup>j</sup>* = *i. Then,* T*n*(*S*) *consists of <sup>n</sup> vertex-disjoint subgraphs:* <sup>T</sup> {1} *<sup>n</sup>* (*S*), <sup>T</sup> {2} *<sup>n</sup>* (*S*), ... , <sup>T</sup> {*n*} *<sup>n</sup>* (*S*)*; each isomorphic to another Cayley graph* T*n*−1(*S* ) = Cay(S*n*<sup>−</sup>1, *S* ) *with S* = *S*\*τ, where τ is the transposition corresponding to the edge incident to the leaf j.*

Readers can refer to [10,21] for the terminology and notation not defined in this paper.
