Formulas counting spanning trees in line graphs and their extensions

TL;DR

This paper derives formulas for counting spanning trees in graphs and their line and middle graphs, based on specific subgraph conditions, revealing new factorization properties related to graph cuts.

Contribution

It introduces a novel formula for counting spanning trees containing a given edge set in graphs with particular subgraph structures, extending to line and middle graphs.

Findings

Derived a formula for $ au_G(M)$ under specific conditions.

Obtained a new factorization property for spanning trees related to graph cuts.

Applied results to line and middle graphs for spanning tree enumeration.

Abstract

For any connected multigraph $G=(V,E)$ and any $M\subseteq E$, if $M$ induces an acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of $G$ whose components are complete graphs, a formula for $\tau_G(M)$ is obtained, where $\tau_G(M)$ is the number of spanning trees in $G$ which contain all edges in $M$. Applying this result, we can easily obtain a formula for the number of spanning trees in the line graph or the middle graph of an arbitrary graph. Applying this result, we also show that for any connected graph $G$ with a clique $U$ which is a cut-set of $G$, the number of spanning trees in $G$ has a factorization which is analogous to a property of the chromatic polynomial of $G$.

Equations

$${\tau }_{K_n}(M)=n^{c-2}\prod _{i=1}^c{n}_i.$$

$${\tau }_G(M_0)=\prod _{i=1}^k|V_i{|}^{|V_i|-2}\sum _{T\in {\mathcal{S}\mathcal{T}}_{G^{\ast }}}\prod _{e\in M_0-E(T)}\left(|V_{a(e)}{|}^{-1}+|V_{b(e)}{|}^{-1}\right),$$

$${\tau }_{L(H)}$$

$${\tau }_{L(H)}=2^{m-n+1}{r}^{m-n-1}{\tau }_H,$$

$$\chi (G,\lambda )=\frac{\chi (G[U\cup S_1],\lambda )\cdot \chi (G[U\cup S_2],\lambda )}{\chi (K_{|U|},\lambda )}.$$

$${\tau }_G(W)={\tau }_{G\star W}(W^{\prime })={\tau }_{G\star W-W_0}(W^{\prime }),$$

$${\tau }_G(W)=|U{|}^{|U|-2+t-n_1-n_2-\cdots -n_t}\prod _{i=1}^t{n}_i,$$

$${\tau }_G(W)={\tau }_{G/E^{\prime }}(E(F/E^{\prime }))={\tau }_{G[U]}(E(F/E^{\prime })),$$

$$n_1,\cdots ,n_t,\underset{|U|-t\text{mbox}numbers}{\underbrace{1,1,\cdots ,1}}.$$

$${\tau }_{G[U]}(F_0)=|U{|}^{|U|-2+t-n_1-\cdots -n_t}\prod _{i=1}^t{n}_i.$$

$${\tau }_{G/E_0}(E(F/E_0))=|U{|}^{|U|-2+t-n_1^{\prime }-n_2^{\prime }-\cdots -n_t^{\prime }}\prod _{i=1}^t{n}_i^{\prime },$$

$${\tau }_{G^{\prime }}(W^{\prime })=|U{|}^{|U|-2+t-n_1-n_2-\cdots -n_t}\prod _{i=1}^t{n}_i,$$

$${\tau }_G(M\cup N)=|U{|}^{|U|-2-|M|}\sum _{T\in {\mathcal{S}\mathcal{T}}_{G\bullet U}(N)}|U{|}^{|E_T(u)|}.$$

$${\tau }_G(M\cup N)=|U{|}^{|U|-2+t-n_1-\cdots -n_t}\prod _{i=1}^t{n}_i,$$

$$|U{|}^{t-n_1-\cdots -n_t}\prod _{i=1}^t{n}_i=U^{-|M|}\sum _{T\in {\mathcal{S}\mathcal{T}}_{G\bullet U}(N)}|U{|}^{|E_T(u)|}.$$

$${\tau }_{G\bullet U}(N)=\prod _{i=1}^t|V(F_i)\cap U|=\prod _{i=1}^t{n}_i.$$

$${\mathcal{S}\mathcal{T}}_G(M\cup N)=\bigcup _{N^{\prime }\in \mathcal{N}}{\mathcal{S}\mathcal{T}}_{G⟨M\cup N^{\prime }\cup E_0⟩}(M\cup N^{\prime }).$$

$${\mathcal{S}\mathcal{T}}_{G\bullet U}(N)=\bigcup _{N^{\prime }\in \mathcal{N}}{\mathcal{S}\mathcal{T}}_{G\bullet U⟨M\cup N^{\prime }⟩}(N^{\prime }).$$

$${\tau }_{G⟨M\cup N^{\prime }\cup E_0⟩}(M\cup N^{\prime })=|U{|}^{|U|-2-|M|}\sum _{T\in {\mathcal{S}\mathcal{T}}_{G\bullet U⟨M\cup N^{\prime }⟩}(N^{\prime })}|U{|}^{|E_T(u)|}.$$

$${\tau }_G(M\cup N)=\prod _{i=1}^k|V_i{|}^{|V_i|-2-|M_i|}\sum _{T\in {\mathcal{S}\mathcal{T}}_{G\bullet U}(N)}\prod _{i=1}^k|V_i{|}^{|E_T(v_i)|},$$

$$\frac{{\tau }_G(M\cup N)}{|V_k{|}^{|V_k|-2-|M_k|}}$$

$$\frac{{\tau }_{(G-B_k)\bullet V_k}((M-B_k)\cup N)}{{\prod }_{i=1}^{k-1}|V_i{|}^{|V_i|-2-|M_i|}}=\sum _{T\in {\mathcal{S}\mathcal{T}}_{(G-B_k)\bullet U}(N\cup (M_k-B_k))}\prod _{i=1}^{k-1}|V_i{|}^{|E_T(v_i)|}.$$

$$\frac{{\tau }_G(M\cup N)}{{\prod }_{i=1}^k|V_i{|}^{|V_i|-2-|M_i|}}$$

$$\{1\le i\le r:\{e_{i,1},e_{i,2}\}\subseteq E(T)\}=I.$$

$${\tau }_G(M)$$

$${\tau }_G(R\cup N)=\prod _{i=1}^k|V_i{|}^{|V_i|-2}\sum _{T\in {\mathcal{S}\mathcal{T}}_{G\bullet U}(N)}\prod _{i=1}^k|V_i{|}^{-|M_i-E(T)|}(1+|V_i|{)}^{|(M_i-R)-E(T)|}.$$

$${\tau }_G(R\cup N)=\sum _{B\subseteq M-R}{\tau }_{G-B}((M-B)\cup N).$$

$${\tau }_G(R\cup N)$$

$$\Phi (R^{\prime })$$

$$\frac{{\tau }_G(R\cup N)}{{\prod }_{i=1}^k|V_i{|}^{|V_i|-2}}$$

Full text

MSC: 05A15, 05C05, 05C30, 05C76

Keywords: Graph, clique, Spanning tree, Cayley’s formula

1 Introduction

The graphs considered in this article are multigraphs without loops. For any graph $G$, let $V(G)$ and $E(G)$ be the vertex set and the edge set of $G$ respectively. For any non-empty $V^{\prime}\subseteq V(G)$, let $G[V^{\prime}]$ denote the subgraph of $G$ induced by $V^{\prime}$, and when $V^{\prime}\neq V(G)$, let $G-V^{\prime}$ be the subgraph $G[V(G)-V^{\prime}]$ (i.e., the subgraph of $G$ obtained by deleting all vertices in $V^{\prime}$). Let $N_{G}(V^{\prime})=\bigcup_{v\in V^{\prime}}N_{G}(v)$, where $N_{G}(v)$ is the set of neighbours of $v$ in $G$, and $N_{G}[V^{\prime}]=V^{\prime}\cup N_{G}(V^{\prime})$. For any $E^{\prime}\subseteq E(G)$, let $G\langle E^{\prime}\rangle$ be the spanning subgraph of $G$ with edge set $E^{\prime}$, let $G[E^{\prime}]$ be the subgraph of $G$ induced by $E^{\prime}$ when $E^{\prime}\neq\emptyset$ (i.e., the graph obtained from $G\langle E^{\prime}\rangle$ by removing all isolated vertices), let $G/E^{\prime}$ be the graph obtained from $G$ by contracting all edges in $E^{\prime}$ and $G-E^{\prime}$ be the subgraph $G\langle E(G)-E^{\prime}\rangle$ (i.e., the graph obtained from $G$ by removing all edges in $E^{\prime}$).

For any graph $G$, let ${\cal ST}_{G}$ be the set of spanning trees of $G$ and let $\tau_{G}=|{\cal ST}_{G}|$. Clearly, $\tau_{G}=0$ if and only if $G$ is disconnected. It is well-known that $\tau(K_{n})=n^{n-2}$, due to Cayley [2], where $K_{n}$ is the complete graph of order $n$. This beautiful formula was extended by Moon [11, 12, 13] (also see Lovász [8, Problem 4 in page 34]) for counting the number of spanning trees $T\in{\cal ST}_{K_{n}}$ which contain all edges of a given forest in $K_{n}$.

For any $M\subseteq E(G)$, let ${\cal ST}_{G}(M)$ be the set of those members $T\in{\cal ST}_{G}$ with $M\subseteq E(T)$ and let $\tau_{G}(M)=|{\cal ST}_{G}(M)|$. Thus ${\cal ST}_{G}(M)\subseteq{\cal ST}_{G}$, where ${\cal ST}_{G}(M)={\cal ST}_{G}$ holds whenever $M$ consists of bridges of $G$. Clearly, $\tau_{G}(M)=0$ if and only if either $G$ is disconnected or $G\langle M\rangle$ contains cycles.

Theorem 1.1** (Lovász [8] and Moon [11, 12, 13])**

For any $M\subseteq E(K_{n})$, if $K_{n}\langle M\rangle$ is a forest with $c$ components whose orders are $n_{1},n_{2},\cdots,n_{c}$, then

[TABLE]

It is natural to consider a suitable extension of Theorem 1.1. In this article, we assume that $G=(V,E)$ is a connected graph, where $V$ can be partitioned into subsets $V_{0},V_{1},\cdots,V_{k}$ and $V_{i}$ is a clique of $G$ (i.e., $G[V_{i}]$ is a complete graph) for all $i=1,2,\cdots,k$. Thus $G[V_{i}]$ has no parallel edges for all $1\leq i\leq n$, although $G$ may have parallel edges. Note that $V_{0}$ may be an empty set and $G[V_{0}]$ may be not complete and may have parallel edges also.

For any $U_{1},U_{2}\subseteq V$, let $E_{G}(U_{1},U_{2})$ denote the set of those edges in $G$ with one end in $U_{1}$ and another end in $U_{2}$, and let $E_{G}(U_{1})=E_{G}(U_{1},V-U_{1})$. In the case that $V_{0}=\emptyset$ and $M_{0}=\bigcup_{1\leq i<j\leq k}E_{G}(V_{i},V_{j})$ is a matching of $G$, an formula for $\tau_{G}(M_{0})$ was obtained in [6, Theorem 3.1]. Let $G^{*}$ be the graph obtained from $G$ by identifying all vertices in each $V_{i}$ as one vertex $v_{i}$ for all $i=1,2,\cdots,k$ and removing all loops. Thus $G^{*}=G/E_{0}$, where $E_{0}=\cup_{1\leq i\leq k}E(G[V_{i}])$.

Theorem 1.2** ([6])**

If $V_{0}=\emptyset$ and $M_{0}$ is a matching of $G$, then

[TABLE]

where $1\leq a(e)<b(e)\leq k$ such that $v_{a(e)}$ and $v_{b(e)}$ are the two ends of $e$ in $G^{*}$ for each $e\in E(T)$.

If $V_{0}=\emptyset$ and $M_{0}$ is a perfect matching of $G$, then $G/M_{0}$ is actually the line graph $L(G^{*})$ of $G^{*}$. Since $\tau_{G}(M_{0})=\tau_{G/M_{0}}$ holds (see Lemma 2.1 (v)), applying Theorem 1.2 yields a relation between $\tau_{L(H)}$ and $\tau_{H}$ for an arbitrary connected graph $H$.

Corollary 1.1** ([6])**

Let $H$ be a connected and loopless graph with vertices $v_{1},v_{2},\cdots,v_{k}$. Then

[TABLE]

where $d(v_{i})$ is the degree of $v_{i}$ in $H$ and $v_{a(e)}$ and $v_{b(e)}$ are the two ends of $e$ in $H$.

For any connected graph $H$, the middle graph $M(H)$ of $H$ is the one obtained from $H$ by subdividing each edge in $H$ exactly once and adding a new edge joining each pair of new vertices $u_{1},u_{2}$ which subdivide a pair of adjacent edges in $H$ (see [4]).

Observe that if $V_{0}=\emptyset$, $M_{0}$ is a matching of $G$ and exactly one vertex in each $V_{i}$ is not incident with $M_{0}$, where $1\leq i\leq k$, then $G/M_{0}$ is actually the middle graph of $G^{*}$, and thus, by the equality $\tau_{G}(M_{0})=\tau_{G/M_{0}}$, a formula for $\tau_{M(H)}$ follows directly from Theorem 1.2.

Remark: The study of a relation between $\tau_{L(H)}$ and $\tau_{H}$ for a connected graph $H$ was started in 1966 when Vahovskii [16] first established such a relation for a $r$-regular graph $H$:

[TABLE]

where $n=|V(H)|$ and $m=|E(H)|$. When $H$ is a graph in which each vertex is of degree $1$ or $r$, where $r$ is a constant, a similar relation between $\tau_{L(H)}$ and $\tau_{H}$ was found by Yan [18] in 2013. When $H$ is an $(a,b)$-semiregular bipartite graph, such a relation was found by Cvetković (see [9, see Theorem 3.9], [10, §5.2], or [15]). Corollary 1.1 was the first result giving a relation between $\tau_{L(H)}$ and $\tau_{H}$ for an arbitrary connected graph $H$, which implies all these known results.

In this article, we will further extend Theorem 1.2. Recall that $V_{0},V_{1},\cdots,V_{k}$ is a partition of $V$, where $V_{i}$ is a clique of $G$ for all $i=1,2,\cdots,k$. Let $M=M_{0}\cup\bigcup_{1\leq i\leq k}E_{G}(V_{0},V_{i})$, where $M_{0}=\bigcup_{1\leq i<j\leq k}E_{G}(V_{i},V_{j})$ is not restricted to a matching of $G$. We will study the set ${\cal ST}_{G}(W)$ for any $W$ with $M\subseteq W\subseteq E(G)$, where $G\langle W\rangle$ is a forest.

In Section 2, we will transform $W,M$ and $G$ so that the study of $\tau_{G}(W)$ can be restricted to the special case that $M_{0}=\emptyset$, each component of $G[M]$ is a star with a center in $V_{0}$ and $W=M\cup N$ for some $N\subseteq E(G[V_{0}])$, as stated in (i), (ii) and (iii) in Page (iii). With these conditions, the structure of $G$ is as shown in Figure 2 of Page (iii). Thus, in Sections 3 and 4, we will study $\tau_{G}(W)$ under these assumptions.

For any $U\subseteq V(G)$, let $G\bullet U$ denote the graph $G/E(G[U])$, i.e., the graph obtained from $G$ by contracting all edges in $G[U]$. In Section 3, we find a relation between $\tau_{G}(M\cup N)$ and $\tau_{G\bullet U}(N)$, where $U$ is a clique of $G$ and $N\subseteq E(G[V-U])$. In Section 4, we will apply the result in Section 3 to obtain a relation between $\tau_{G}(M\cup N)$ and $\tau_{G\bullet U}(N)$, where $U$ is the union of $k$ disjoint cliques $V_{1},V_{2},\cdots,V_{k}$ and $N\subseteq E(G[V-U])$.

In Section 5, as an application of the results in Section 4, we find a formula for $\tau_{G}$ when $E(G)$ can be partitioned into subsets $E_{1},E_{2},\cdots,E_{k}$ such that each $G[E_{i}]$ is a clique in $G$. The middle graph and the line graph of any connected graph $H$ are examples of such graphs. Applying the results in Section 4, one can easily deduce formulas for $\tau_{M(H)}$ and $\tau_{L(H)}$ for any given connected graph $H$.

It is well known that if $U$ is a clique of $G$ and $S_{1},S_{2}$ is a partition of $V-U$ such that $E_{G}(S_{1},S_{2})=\emptyset$, then the following equality for the chromatic polynomial $\chi(G,\lambda)$ of $G$ holds (see [5, 14, 19]):

[TABLE]

Section 6 shows that $\tau_{G}$ has a similar result as (1.5) when $N_{G}[S_{1}]\cap N_{G}[S_{2}]=\emptyset$ holds.

2

It suffices to study $\tau_{G}(W)$ for a special case

Let $G=(V,E)$ be a connected graph whose vertex set has a partition $V_{0},V_{1},\cdots,V_{k}$, where each $V_{i}$ is a clique for all $i=1,2,\cdots,k$. In this section, we will show that for any set $W$ with $\bigcup_{0\leq i\leq k}E_{G}(V_{i},V_{j})\subseteq W\subseteq E$, the study of $\tau_{G}(W)$ can be transformed to the special case that each component of $G[W]$ is a star and $E_{G}(V_{i},V_{j})=\emptyset$ for all $i,j$ with $1\leq i<j\leq k$.

2.1

$\tau_{G}(W)=\tau_{G\star W}(W^{\prime})$ holds for $W^{\prime}=E(G\star W)-E(G)$

For any $E^{\prime}\subseteq E$, one can easily prove the following basic properties on ${\cal ST}_{G}(E^{\prime})$.

Lemma 2.1

Let $E^{\prime}\subseteq E$ and $e\in E$. Then

1. (i)

if $G\langle E^{\prime}\rangle$ contains cycles, then ${\cal ST}_{G}(E^{\prime})=\emptyset$; 2. (ii)

if $e\notin E^{\prime}$ and $e$ is a loop, ${\cal ST}_{G}(E^{\prime})={\cal ST}_{G-e}(E^{\prime})$; 3. (iii)

if $e\notin E^{\prime}$ and $e$ is parallel to an edge in $E^{\prime}$, then ${\cal ST}_{G}(E^{\prime})={\cal ST}_{G-e}(E^{\prime})$; 4. (iv)

if $e\in E^{\prime}$ and $e$ is not a loop, then $\tau_{G}(E^{\prime})=\tau_{G/e}(E^{\prime}-\{e\})$; 5. (v)

if $G\langle E^{\prime}\rangle$ is a forest, $\tau_{G}(E^{\prime})=\tau_{G/E^{\prime}}$; 6. (vi)

if $e\notin E^{\prime}$ and $G[E^{\prime}\cup\{e\}]$ has a cycle containing $e$, then ${\cal ST}_{G}(E^{\prime})={\cal ST}_{G-e}(E^{\prime})$.

Proof.  (i) and (iii) follow directly from the definition of ${\cal ST}_{G}(E^{\prime})$. Both (ii) and (vi) follow from the fact that $e$ is not contained in any tree $T\in{\cal ST}_{G}(E^{\prime})$. (iv) follows from the fact that $T\in{\cal ST}_{G}(E^{\prime})$ if and only if $T/e\in{\cal ST}_{G/e}(E^{\prime}-\{e\})$. (v) follows from (iv) directly. $\Box$

For any $W\subseteq E$, let $G\star W$ denote the graph obtained from $G$ by adding a new vertex $w_{i}$ and new edges joining $w_{i}$ to all vertices in $W_{i}$ for all $i=1,2,\cdots,r$, where $W_{1},\cdots,W_{r}$ are the components of $G[W]$. An example of $G\star W$ is shown in Figure 1, where $G[W]$ has three components. Thus, $V(G\star W)=V(G)\cup\{w_{1},w_{2},\cdots,w_{r}\}$ and $E(G\star W)=E(G)\cup\bigcup_{1\leq i\leq r}E_{G\star W}(w_{i})$. Also note that $\{w_{1},w_{2},\cdots,w_{r}\}$ is an independent set in $G\star W$.

Lemma 2.2

Let $W\subseteq E$ such that $G[W]$ is a forest. For any $W_{0}\subseteq W$,

[TABLE]

where $W^{\prime}=E(G\star W)-E(G)$.

Proof.  Observe that $\tau_{G\star W}(W^{\prime})=\tau_{G\star W-W_{0}}(W^{\prime})$ follows from Lemma 2.1 (vi) directly, while $\tau_{G}(W)=\tau_{G\star W}(W^{\prime})$ follows from Lemma 2.1 (v) and the fact that the two graphs obtained respectively from $G/W$ and $(G\star W)/W^{\prime}$ by removing their loops are isomorphic.

Thus the result follows. $\Box$

2.2 Transformed to a special case

Recall that $V_{0},V_{1},\cdots,V_{k}$ is a partition of $V$ such that each $V_{i}$ is a clique of $G$ for all $i=1,2,\cdots,k$. Let $M=\bigcup_{0\leq i<j\leq k}E_{G}(V_{i},V_{j})$ and $W$ be a subset of $E$ with $M\subseteq W$ such that $G[W]$ is a forest. By Lemma 2.2, we get the following conclusion.

Lemma 2.3

Let $G^{\prime}$ denote the graph $G\star W-M$, $V^{\prime}_{0}=V_{0}\cup(V(G\star W)-V(G))$ and $W^{\prime}=E(G\star W)-E(G)$. The following properties hold:

1. (i)

$\tau_{G}(W)=\tau_{G^{\prime}}(W^{\prime})$; 2. (ii)

$V^{\prime}_{0},V_{1},\cdots,V_{k}$* is a partition of $V(G^{\prime})$, where $V_{i}$ is a clique of $G^{\prime}$ for all $i=1,2,\cdots,k$;* 3. (iii)

$E_{G^{\prime}}(V_{i},V_{j})=\emptyset$* for all $i,j$ with $1\leq i<j\leq k$;* 4. (iv)

each component of $G^{\prime}[W^{\prime}]$ is a star with a center in $V(G^{\prime})-V(G)\subseteq V^{\prime}_{0}$.

Lemma 2.3 (i) follows from Lemma 2.2 while Lemma 2.3 (ii)-(iv) follow directly from the definitions of $G^{\prime}$ and $W^{\prime}$. By Lemma 2.3, the study of $\tau_{G}(W)$ can be restricted to the special case that $V(G)$ has a partition $V_{0},V_{1},\cdots,V_{k}$ satisfying the following conditions:

1. (i)

$V_{i}$ is a clique for all $i=1,2,\cdots,k$ and $E_{G}(V_{i},V_{j})=\emptyset$ holds for each pair of $i,j$ with $1\leq i<j\leq k$; 2. (ii)

for $M=\bigcup_{0\leq i\leq k}E_{G}(V_{0},V_{i})$, each component of $G[M]$ is a star with a center in $V_{0}$ (i.e., $d_{G}(u)\leq|V_{i}|$ holds for each $u\in V(G)-V_{0}$); 3. (iii)

$W=M\cup N$ for some $N\subseteq E(G[V_{0}])$.

When the above three conditions holds, $G$ has its structure as shown in Figure 2.

3

Contracting a clique $U$

Let $U$ be a clique of a connected graph $G=(V,E)$. In Subsection 3.1, we will deduce a formula for $\tau_{G}(W)$ in the case that $G[W]$ is a forest, where $W=E-E(G[U])$. Let $G\bullet U$ denote the graph $G/G[U]$. In Subsection 3.2, we will give a relation between $\tau_{G}(M\cup N)$ and $\tau_{G\bullet U}(N)$, where $M=E_{G}(U)$ and $N\subseteq E(G-U)$, under the condition that each component of $G[M]$ is a star with a center in $V-U$.

3.1

When $G-E(G[U])$ is a forest

Note that $G-E(G[U])$ is a forest if and only if $G[W]$ is a forest, where $W=E-E(G[U])$. When $G[W]$ is a forest, applying Theorem 1.1, we get a formula for $\tau_{G}(W)$ below.

Proposition 3.1

Let $U$ be a clique of $G$ with $U\neq V$ and $W=E-E(G[U])$. If $F=G[W]$ is a forest with components $F_{1},F_{2},\cdots,F_{t}$, then

[TABLE]

where $t$ is the number of components of $F$ and $n_{i}=|V(F_{i})\cap U|$ for $i=1,2,\cdots,t$.

Proof.  Note that for any $i=1,2,\cdots,t$, $|E(F_{i})|\geq n_{i}$, and $|E(F_{i})|=n_{i}$ if and only if $F_{i}$ is a star with a center in $V-U$ and $E(F_{i})\subseteq E_{G}(U)$.

We shall prove this result by the following claims.

Claim 1: (3.1) holds when each $F_{i}$ is a star with a center at $V-U$ and $E(F_{i})\subseteq E_{G}(U)$.

Assume that each $F_{i}$ is a star with a center at $V-U$ and $E(F_{i})\subseteq E_{G}(U)$. Then $G-U$ is an empty graph, $V=U\cup N_{G}(U)$ and $F$ is the bipartite graph $G[E_{G}(U)]$.

Let $E^{\prime}=\{e_{1},e_{2},\cdots,e_{t}\}$, where $e_{i}$ is an edge in $F_{i}$. Applying Lemma 2.1 (iv) and (iii) repeatedly, we have

[TABLE]

where $F/E^{\prime}$ is considered as a subforest of $G[U]$ whose components’s vertex sets are $U\cap V(F_{i})$ for $i=1,2,\cdots,t$. Note that $F_{0}=G\langle E(F/E^{\prime})\rangle$ is a spanning forest of $G[U]$ with $|U|-n_{1}-n_{2}-\cdots-n_{t}+t$ components with the following orders:

[TABLE]

By Theorem 1.1, we have

[TABLE]

Hence Claim 1 holds.

Claim 2: (3.1) holds when each $F_{i}$ is a star with a center at $V-U$.

Assume that each $F_{i}$ is a star with a center at $V-U$, as shown in Figure 3.

If $G-U$ is an independent set of $G$, then each $F_{i}$ is a star with a center in $V-U$ and $E(F_{i})\subseteq E_{G}(U)$, implying that the claim holds by Claim 1.

Now assume that $V-U$ is not independent in $G$, i.e., $E_{0}=E(G-U)\neq\emptyset$. Since each component $F_{i}$ is a star with a center in $V-U$, each edge $e\in E_{0}$ is incident with two vertices in $V-U$ one of which is an end-vertex. Thus $U$ is still a clique of $G/E_{0}$ and $F/E_{0}$ is a forest with $t$ components $F^{\prime}_{1},F^{\prime}_{2},\cdots,F^{\prime}_{t}$ each of which is a star with a center in $V(G/E_{0})-U$ and each edge in $E(F/E_{0})$ is incident with some vertex in $U$, where $F^{\prime}_{i}=F_{i}/(E_{0}\cap E(F_{i}))$. By Claim 1, the result holds for $G/E_{0}$, i.e.,

[TABLE]

where $n^{\prime}_{i}=|U\cap V(F^{\prime}_{i})|$. Clearly $n^{\prime}_{i}=|U\cap V(F^{\prime}_{i})|=|U\cap V(F_{i})|=n_{i}$. Applying Lemma 2.1 (iv) repeatedly, we have $\tau_{G}(E(F))=\tau_{G/E_{0}}(E(F/E_{0}))$. Thus Claim 2 holds.

Claim 3: (3.1) holds whenever $F$ is a forest.

Let $W=E(F)=E(G-G[U])$ and $G^{\prime}=G\star W-W$. Observe that $U$ is a clique of $G^{\prime}$ and $G^{\prime}[E(G^{\prime})-E(G^{\prime}[U])]$ is a forest with $t$ components $F^{\prime}_{1},\cdots,F_{t}^{\prime}$ each of which is a star with a center in $V(G^{\prime})-U$ such that $V(F^{\prime}_{i})\cap U=V(F_{i})\cap U$ holds for all $i=1,2,\cdots,t$. By Claim 2, the result holds for $G^{\prime}$, i.e.,

[TABLE]

where $W^{\prime}=E(G\star W)-W$. By Lemma 2.2, we have $\tau_{G}(W)=\tau_{G\star W-W}(W^{\prime})$.

Thus Claim 3 holds and the result is proved. $\Box$

Remark: For any forest $M$ in $K_{n}$ with components $M_{1},\cdots,M_{t}$, let $G=K_{n}\star M$. By Lemma 2.2, $\tau_{K_{n}}(M)=\tau_{G}(W)$, where $W=E(G)-E(K_{n})$. Note that each component $F_{i}$ of $G[W]$ is a star with a center in $V(G)-U$ and $E(F_{i})\subseteq E_{G}(U)$, where $U=V(K_{n})$ and $V(F_{i})\cap U=V(M_{i})$. Theorem 1.1 corresponds to Proposition 3.1 for the case that each component $F_{i}$ of $F$ is a star with a center in $V-U$ and $E(F_{i})\subseteq E_{G}(U)$.

3.2

Relation between $\tau_{G}(M\cup N)$ and $\tau_{G\bullet U}(N)$

In this subsection, we assume that $U$ is a clique of $G$ and each component of $G[E_{G}(U)]$ is a star with a center in $V-U$. Note that each component of $G[E_{G}(U)]$ is a star with a center in $V-U$ if and only if each vertex in $U$ is incident with at most one edge in $E_{G}(U)$.

Let $u$ be the new vertex in $G\bullet U$ created after contracting all edges in $E(G[U])$. So the vertex set of $G\bullet U$ is $(V-U)\cup\{u\}$. Note that $G\bullet U$ may have parallel edges incident with vertex $u$, as all edges in $E_{G}(U)=E_{G}(U,V-U)$ are the edges in $G\bullet U$ incident with $u$. For each $v\in V-U$, the number of parallel edges in $G\bullet U$ joining $u$ and $v$ is equal to $|N_{G}(v)\cap U|$.

Now we are going to establish the main result in this section.

Theorem 3.1

Let $M=E_{G}(U)$ and $N\subseteq E(G-U)$. If each component of $G[M]$ is a star with a center in $V-U$, then

[TABLE]

Proof.  By the given condition on $M$, $G[M\cup N]$ contains cycles if and only if $G\bullet U[N]$ contains cycles, implying that (3.7) holds whenever $G[M\cup N]$ contains cycles. Thus, it suffices to consider the case that $G[M\cup N]$ is a forest. We will prove (3.7) by completing the following claims.

Claim 1: (3.7) holds if $G-E(G[U])$ is a forest and $N=E(G-U)$.

Assume that $N=E(G-U)$ and $G[M\cup N]$ is a forest with components $F_{1},F_{2},\cdots,F_{t}$, as shown in Figure 4. By Proposition 3.1,

[TABLE]

where $n_{i}=|V(F_{i})\cap U|$ for $i=1,2,\cdots,t$, implying that (3.7) holds if and only if the following equality holds:

[TABLE]

By the given condition, each vertex in $U$ is incident with at most one edge in $M$. Since $n_{1}+n_{2}+\cdots+n_{t}$ is the number of vertices in $U$ which are incident with edges in $M$, we have $n_{1}+n_{2}+\cdots+n_{t}=|M|$. For any $T\in{\cal ST}_{G\bullet U}(N)$, we have $|E(T)\cap E_{G}(U,V(F_{i})-U)|=1$ for all $i=1,2,\cdots,t$, implying that $|E_{T}(u)|=t$.

It remains to show that $\tau_{G\bullet U}(N)=\prod_{i=1}^{t}n_{i}$. Let $T\in{\cal ST}_{G\bullet U}(N)$. Observe that $T-u$ is actually the graph $G-U$, which consists of $t$ components $F_{i}-V(F_{i})\cap U$ for $i=1,2,\cdots,t$. Also note that $T$ contains exactly $t$ edges $e_{1},e_{2},\cdots,e_{t}$, where each $e_{i}$ with $u$ and some vertex in $F_{i}-V(F_{i})\cap U$ for $i=1,2,\cdots,t$. Observe that each $e_{i}$ can be any one of the edges in the set $M\cap E(F_{i})$ whose size is exactly $|V(F_{i})\cap M|=n_{i}$. Hence

[TABLE]

Thus (3.9) holds and Claim 1 follows.

Claim 2: (3.7) holds for any $N\subseteq E(G-U)$ such that $G[M\cup N]$ is a forest.

For any $T\in{\cal ST}_{G}(M\cup N)$, $T-U$ is a forest with $N\subseteq E(T-U)$. Let $E_{0}=E(G(U))$ and let ${\cal N}$ be the family of those subsets $N^{\prime}$ of $E(G-U)$ with $N\subseteq N^{\prime}$ such that $G\langle N^{\prime}\rangle$ is a forest and $G\langle M\cup N^{\prime}\cup E_{0}\rangle$ is connected.

Clearly, ${\cal ST}_{G\langle M\cup N_{1}\cup E_{0}\rangle}(M\cup N_{1})$ and ${\cal ST}_{G\langle M\cup N_{2}\cup E_{0}\rangle}(M\cup N_{2})$ are disjoint for any pair of distinct members $N_{1},N_{2}\in{\cal N}$, and

[TABLE]

Similarly, for any pair of distinct members $N_{1},N_{2}\in{\cal N}$, ${\cal ST}_{G\bullet U\langle M\cup N_{1}\rangle}(N_{1})$ and ${\cal ST}_{G\bullet U\langle M\cup N_{2}\rangle}(N_{2})$ are disjoint, and

[TABLE]

By Claim 1, the following identity holds for any $N^{\prime}\in{\cal N}$:

[TABLE]

Thus Claim 2 follows from (3.11), (3.12) and (3.13). $\Box$

4

When $V_{1},V_{2},\cdots,V_{k}$ are disjoint cliques of $G$

In this section, we always assume that $G=(V,E)$ is a connected and loopless multigraph, where $V$ is partitioned into non-empty subsets $V_{0},V_{1},\cdots,V_{k}$ satisfying the following conditions:

1. (i)

$V_{i}$ is a clique for all $i=1,2,\cdots,k$; 2. (ii)

$E_{G}(V_{i},V_{j})=\emptyset$ for each pair $i,j$ with $1\leq i<j\leq k$; 3. (iii)

each component of $G[M]$ is a star with a center in $V_{0}$, where $M=\bigcup\limits_{1\leq i\leq k}E_{G}(V_{0},V_{i})$.

The structure of $G$ under conditions (i), (ii) and (iii) above is as shown in Figure 5(a). Note that condition (iii) above is equivalent to that each vertex in $V_{i}$ is incident with at most one edge in $M$ for all $i=1,2,\cdots,k$. All parallel edges of $G$ must be in the subgraph $G[V_{0}]$.

Let $M_{i}=E_{G}(V_{i},V_{0})$ for all $i=1,2,\cdots,k$. Then $M=E_{G}(V_{0})=\bigcup\limits_{1\leq i\leq k}M_{i}$. In this section, our main purpose is to apply Theorem 3.1 to find an expression for $\tau_{G}(M\cup N)$ for any $N\subseteq E(G[V_{0}])$. Applying this result, we are able to get an expression of $\tau_{G}(R\cup N)$ for any $R\subseteq M$.

4.1 $\tau_{G}(M\cup N)$ for $N\subseteq E(G[V_{0}])$

Let $U=V_{1}\cup\cdots\cup V_{k}$. Recall that $G\bullet U$ is defined to be the graph $G/E(G[U])$. As $G[U]$ has $k$ components $G[V_{1}],G[V_{2}],\cdots,G[V_{k}]$, $G\bullet U$ can be obtained from $G$ by removing all edges in $G[U]$ and identifying all vertices in each $V_{i}$ as one vertex, denoted by $v_{i}$, for $i=1,2,\cdots,k$, as shown in Figure 5 (b). Thus $V(G\bullet U)=V_{0}\cup\{v_{1},v_{2},\cdots,v_{k}\}$ and $E(G\bullet U)=E(G)-\bigcup_{1\leq i\leq k}E(G[V_{i}])=M\cup E(G[V_{0}])$.

Theorem 4.1

For any $N\subseteq E(G[V_{0}])$,

[TABLE]

where $M_{i}=E_{G}(V_{i},V_{0})$ for $i=1,2,\cdots,k$ and $M=M_{1}\cup\cdots\cup M_{k}$.

Proof.  If $k=1$, the result follows directly from Theorem 3.1. Assume that the result holds for $k<n$, where $n\geq 2$. Now consider the case that $k=n$.

By Theorem 3.1,

[TABLE]

where the last equality follows from the fact that $T\in{\cal ST}_{G\bullet V_{k}}((M-M_{k})\cup N)$ with $E_{T}(v_{k})=M_{k}-B_{k}$ if and only if $T\in{\cal ST}_{(G-B_{k})\bullet V_{k}}((M-B_{k})\cup N)$, as $M-B_{k}=(M-M_{k})\cup(M_{k}-B_{k})$.

For any $B_{k}\subseteq M_{k}$, we have $M-B_{k}=M_{1}\cup\cdots\cup M_{k-1}\cup(M_{k}-B_{k})$. By the inductive assumption,

[TABLE]

By (4.2) and (4.3), we have

[TABLE]

$\Box$

Now we give a proof of Theorem 1.2 by applying Theorem 4.1 directly.

Proof of Theorem 1.2. Assume that $G=(V,E)$ is a graph satisfying the conditions assumed in the beginning of this section, $V_{0}$ is an independent set of $G$ and each component of $G[M]$ is a star of size $2$.

Let $V_{0}=\{w_{1},w_{2},\cdots,w_{r}\}$. Then $G[M]$ consists of exactly $r$ components $S_{1},S_{2},\cdots,S_{r}$ which are stars of size $2$ with centers $w_{1},w_{2},\cdots,w_{r}$ respectively. Let $e_{i,1}$ and $e_{i,2}$ denote the two edges in $S_{i}$.

Clearly, $\{e_{i,1},e_{i,2}\}\cap E(T)\neq\emptyset$ holds for any $1\leq i\leq r$ and any $T\in{\cal ST}_{G\bullet U}$. For any $I\subseteq\{1,2,\cdots,r\}$, let ${\cal ST}_{G\bullet U}^{I}$ be the set of members $T\in{\cal ST}_{G\bullet U}$ such that

[TABLE]

Observe that ${\cal ST}_{G\bullet U}^{I}\neq\emptyset$ if and only if the edge set $\{e_{i,1},e_{i,2}:i\in I\}$ induces a spanning tree of the subgraph $(G\bullet U)-\{w_{j}:1\leq j\leq r,j\notin I\}$. When ${\cal ST}_{G\bullet U}^{I}\neq\emptyset$, there are exactly $2^{r-|I|}$ members in ${\cal ST}_{G\bullet U}^{I}$ and for each $T\in{\cal ST}_{G\bullet U}^{I}$, $|E(T)\cap\{e_{i,1},e_{i,2}\}|=1$ holds for all $i\in\{1,2,\cdots,r\}-I$.

Let $G^{\prime}$ be the graph obtained from $G$ by contracting exactly one edge in each $S_{i}$ for all $i=1,2,\cdots,r$, and let $M^{\prime}=M\cap E(G^{\prime})$. By Lemma 2.1 (iv), $\tau_{G^{\prime}}(M^{\prime})=\tau_{G}(M)$.

For any $e\in M$, let $l(e)=i$ such that $e$ is incident with a vertex in $V_{i}$. By Theorem 4.1,

[TABLE]

where $G^{*}$ is the graph obtained from $G^{\prime}$ by identifying all vertices in each $V_{i}$ as a vertex, denoted by $v_{i}$, and removing all loops, and $a(e)$ and $b(e)$ are numbers in $\{1,2,\cdots,k\}$ such that $v_{a(e)}$ and $v_{b(e)}$ are the two ends of $e$ in $G^{*}$ which correspond to $V_{a(e)}$ and $V_{b(e)}$.

Since $|{\cal ST}_{G^{\prime}}(M^{\prime})|=\tau_{G}(M)$, Theorem 1.2 is proven by (4.5). $\Box$

4.2 $\tau_{G}(R\cap N)$ for $R\subseteq M$

and $N\subseteq E(G-U)$

In this subsection, we will find an expression for $\tau_{G}(R\cup N)$ for any $R\subseteq E_{G}(U)$ and $N\subseteq E(G-U)$.

Theorem 4.2

For any $R\subseteq M$ and $N\subseteq E(G-U)$,

[TABLE]

Proof.  Let $R$ be a fixed subset of $M$. Note that $T\in{\cal ST}_{G}(R\cup N)$ if and only if $T\in{\cal ST}_{G-B}((M-B)\cup N)$ for some $B$ with $B\subseteq M-R$. Obviously, for distinct subsets $B_{1},B_{2}$ of $M-R$, ${\cal ST}_{G-B_{1}}((M-B_{1})\cup N)$ and ${\cal ST}_{G-B_{2}}((M-B_{2})\cup N)$ are disjoint. Thus,

[TABLE]

Then, by Theorem 4.1,

[TABLE]

where

[TABLE]

By (4.9) and (4.8), we have

[TABLE]

Thus we can verify that the result holds. $\Box$

When $R=\emptyset$, a direct application of Theorem 4.2 gives an expression for $\tau_{G}(N)$.

Corollary 4.1

For any $N\subseteq E(G-U)$,

[TABLE]

4.3

Simplify the expression of (4.6)

Note that (4.6) can be changed to

[TABLE]

For any $R\subseteq M$, let $\omega$ be the mapping from $E(G\bullet U)$ (i.e., $M\cup E(G-U)$) to ${\mathbb{N}}=\{1,2,3,\cdots\}$ defined below:

[TABLE]

Then (4.6) can be expressed as

[TABLE]

Let $w_{1},w_{2},\cdots,w_{r}$ be the vertices in the set $N_{G}(U)-U$. These vertices are actually centers of the components of $G[M]$, as each component of $G[M]$ is a star.

As there may be more than one edge in $E_{G\bullet U}(v_{i},w_{j})\cap R$ or $E_{G\bullet U}(v_{i},w_{j})-R$ for $1\leq i\leq k$ and $1\leq j\leq r$, (4.14) can be further simplified.

Given $R\subseteq M$, let $G\circ_{R}U$ denote the graph obtained from $G\bullet U$ by removing $|E_{G\bullet U}(v_{i},w_{j})\cap R|-1$ edges in the set $E_{G\bullet U}(v_{i},w_{j})\cap R$ whenever $|E_{G\bullet U}(v_{i},w_{j})\cap R|\geq 2$ and $|E_{G\bullet U}(v_{i},w_{j})-R|-1$ edges in the set $E_{G\bullet U}(v_{i},w_{j})-R$ whenever $|E_{G\bullet U}(v_{i},w_{j})-R|\geq 2$ for each pair $i,j:1\leq i\leq k$ and $1\leq j\leq r$. Thus, in the graph $G\circ_{R}U$, there are at most two parallel edges joining each pair of vertices $v_{i}$ and $w_{j}$. If this case happens, then exactly one of the two edges joining $v_{i}$ and $w_{j}$ is contained in $R$.

Let $\omega^{\prime}_{R}$ be the mapping from $E(G\circ_{R}U)$ to ${\mathbb{N}}=\{1,2,3,\cdots\}$ defined below:

[TABLE]

where $1\leq i\leq k$ and $1\leq j\leq r$. Then (4.14) can be replaced by the following expression:

[TABLE]

5

When $E_{1},E_{2},\cdots,E_{k}$ is a partition of $E$ such that each $G[E_{i}]$ is a complete graph

5.1 $\tau_{G}=|{\cal ST}_{G\diamond{\cal S}}|$ holds

for a graph $G\diamond{\cal S}$

Let $v$ be any vertex in $G$ and $E_{0}\subseteq E_{G}(v)$. Let $G_{v\triangleleft E_{0}}$ denote the graph obtained from $G-(E_{G}(v)-E_{0})$ by adding a new vertex $v^{\prime}$ and a new edge joining $v$ and $v^{\prime}$ and finally changing the end $v$ of all edges in $E_{G}(v)-E_{0}$ to $v^{\prime}$, as shown in Figure 6. Clearly $G\triangleleft E_{0}/e^{\prime}\cong G$, where $e^{\prime}=vv^{\prime}$ is the only edge in $E(G\triangleleft E_{0})-E(G)$. By Lemma 2.1 (iv), $\tau_{G}(W)=|{\cal ST}_{G_{v\triangleleft E_{0}}}(W\cup e^{\prime})|$ holds.

For any subgraph $G_{0}$ of $G$, let $G\diamond G_{0}$ be the graph below:

[TABLE]

where $v_{1},v_{2},\cdots,v_{r}$ are those vertices in $G_{0}$ with $E_{G}(v_{i})\neq E_{G_{0}}(v_{i})$ and $E_{i}=E_{G_{0}}(v_{i})$. Clearly, $G_{0}$ is the subgraph of $G\diamond G_{0}$ induced by $V(G_{0})$ and the edges in $E(G\diamond G_{0})-E(G)$ form a matching of $G\diamond G_{0}$. An example is shown in Figure 7, where $G_{0}=G[\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}\}]-\{v_{1}v_{6},v_{4}v_{5}\}$ and the new edges in $G\diamond G_{0}$ are expressed by lines. If $G\cong K_{5}$ and $G_{0}$ is a $5$-cycle, then $G\diamond G_{0}$ is the Petersen graph.

Note that $E(G\diamond G_{0})-E(G)$ is a matching of $G\diamond G_{0}$. Since $G$ is actually the graph obtained from $G\diamond G_{0}$ by contracting all edges in $E(G\diamond G_{0})-E(G)$, applying Lemma 2.1 (iv) repeatedly on all edges in $E(G\diamond G_{0})-E(G)$ implies the following result.

Lemma 5.1

Let $G_{0}$ be a subgraph of $G$ and $M=E(G\diamond G_{0})-E(G)$. Then, for any $N\subseteq E(G)$,

[TABLE]

For a family ${\cal S}=\{E_{1},E_{2},\cdots,E_{k}\}$ of pairwise disjoint subsets of $E(G)$, let $G\diamond{\cal S}$ denote the following graph obtained by a sequence of $\diamond$-operations on subgraphs $G[E_{1}],G[E_{2}],\cdots,G[E_{k}]$:

[TABLE]

Note that $G\diamond{\cal S}$ is irrelevant to the order of $E_{1},E_{2},\cdots,E_{k}$ and $(G\diamond{\cal S})/W\cong G$, where $W=E(G\diamond{\cal S})-E(G)$. An example of $G\diamond{\cal S}$ is shown in Figure 8, where ${\cal S}=\{E_{1},E_{2},E_{3},E_{4}\}$, $E_{1}=E(G[v_{1},v_{5},v_{6}])$, $E_{2}=E(G[v_{2},v_{5},v_{7},v_{9}])$, $E_{3}=E(G[v_{3},v_{7},v_{8}])$ and $E_{4}=E(G[v_{4},v_{6},v_{8},v_{9}])$.

Assume that $V(G)=\{v_{1},v_{2},\cdots,v_{n}\}$. By the above definition, if ${\cal S}=\{E_{1},E_{2},\cdots,E_{k}\}$ is a partition of $E(G)$, then $G\diamond{\cal S}$ is actually the graph with vertex set:

[TABLE]

where $V^{\prime}=V(G)-\{v_{i}\in V(G):\exists j,N_{G}(v_{i})=E_{j}\}$ and $V_{j}=\{v_{i,j}:v_{i}\in V(G[E_{j}])\}$, and edge set:

[TABLE]

where each $E^{\prime}_{j}$ is a copy of $E_{j}$ by changing the ends $v_{s}$ and $v_{t}$ of each edge $e$ in $E_{j}$ to $v_{s,j}$ and $v_{t,j}$. As each edge in $G$ has exactly one copy in $G\diamond{\cal S}$, $E(G\diamond{\cal S})$ is also considered as the union of $E(G)$ and $\bigcup_{v_{i}\in V^{\prime}}\{w_{i}v_{i,j}:E_{G}(v_{i})\cap E_{j}\neq\emptyset,1\leq j\leq k\}$. An example for the labels of vertices and edges in $G\diamond{\cal S}$ is given in Figure 8.

Some basic facts on $G\diamond{\cal S}$ follow directly.

Lemma 5.2

Let $M=E(G\diamond{\cal S})-E(G)$. Then

1. (i)

$|V(G\diamond{\cal S})-V(G)|=|V^{\prime}|$; 2. (ii)

the component number of $(G\diamond{\cal S})[M]$ is equal to $|V^{\prime}|$; 3. (iii)

each component of $(G\diamond{\cal S})[M]$ is a star $S_{j}$ with its center $w_{j}\in V(G\diamond{\cal S})-V(G)$ and its size equal to the number of different sets $E_{i}$ with $E_{G}(v_{j})\cap E_{i}\neq\emptyset$; 4. (iv)

$\{w_{i}:v_{i}\in V^{\prime}\}$* is an independent set in $G\diamond{\cal S}$ and its removal from $G\diamond{\cal S}$ results in $k$ components isomorphic to $G[E_{1}],G[E_{2}],\cdots,G[E_{k}]$ respectively;* 5. (v)

$G\diamond{\cal S}[V_{j}]\cong G[E_{j}]$* for each $j$;* 6. (vi)

$E_{G\diamond{\cal S}}(V_{j_{1}},V_{j_{2}})=\emptyset$* for all $1\leq j_{1}<j_{2}\leq k$.*

Applying Lemma 5.1, we get the following conclusion.

Lemma 5.3

For any partition ${\cal S}=\{E_{1},E_{2},\cdots,E_{k}\}$ of $E(G)$ and any $N\subseteq E(G)$, we have

[TABLE]

where $M=E(G\diamond{\cal S})-E(G)$.

If $G[E_{i}]$ is a complete graph in $G$ for all $i=1,2,\cdots,k$, applying Lemma 5.3 and Theorem 4.1 gets the following expression on $\tau_{G}$.

Theorem 5.1

Assume that ${\cal S}=\{E_{1},E_{2},\cdots,E_{k}\}$ is a partition of $E(G)$ such that $G[E_{i}]$ is a complete graph for all $i=1,2,\cdots,k$. Then

[TABLE]

where $n_{i}=|V(G[E_{i}])|$, $n^{\prime}_{i}$ is the size of the set $\{v_{s}\in V(G[E_{i}]):E_{G}(v_{s})\not\subseteq E_{i}\}$ and $v^{\prime}_{i}$ is the new vertex in $(G\diamond{\cal S})/E(G)$ produced by contracting $V_{i}$ in $G\diamond{\cal S}$.

Note that $(G\diamond{\cal S})/E(G)$ is the bipartite graph with a bipartition $\{w_{i}:v_{i}\in V^{\prime}\}$ and $\{v^{\prime}_{j}:1\leq j\leq k\}$ and edge set $\{w_{i}v^{\prime}_{j}:E_{G}(v_{i})\cap E_{j}\neq\emptyset\}$.

5.2 Application

Obviously Theorem 5.1 can be applied to the graph in Figure 8 (a). Actually this graph is the middle graph of $K_{4}-e$ (i.e., the graph with one edge removed from $K_{4}$). For a graph $H$ with vertex set $\{u_{1},u_{2},\cdots,u_{k}\}$, the middle graph of $H$, denoted by $M(H)$, is the graph obtained from its line graph $L(H)$ and the empty graph $H-E(H)$ by adding edges joining each vertex $u_{i}$ in $H-E(H)$ to all those vertices in $L(H)$ which correspond to edges in the set $E_{H}(u_{i})$.

Applying generalized Wye-Delta transform and Delta-Wye transform, Yan [17] gave a relation between ${\cal ST}_{M(H)}$ and ${\cal ST}_{S(H)}$, where $S(H)$ is the graph obtained from $H$ by subdividing each edge in $H$ exactly once. Such a relation actually follows from Theorem 5.1 directly. Observe that the edge set of $M(H)$ has a partition ${\cal S}=\{E_{1},E_{2},\cdots,E_{k}\}$, where each $E_{i}$ is the set of edges in the subgraph of $M(H)$ induced by $\{u_{i}\}\cup N_{M(H)}(u_{i})$. Clearly each $M(H)[E_{i}]$ is a complete graph of order $d_{H}(u_{i})+1$ and contains exactly $d_{H}(u_{i})$ vertices $u$ such $E_{M(H)}(u)\not\subseteq M(H)[E_{i}]$. Also note that $M(H)\diamond{\cal S}/E(M(H))$ is actually the graph $S(H)$. Thus, applying Theorem 5.1, we have

[TABLE]

Similarly, a relation between ${\cal ST}_{L(H)}$ and ${\cal ST}_{S(H)}$ can be obtained:

[TABLE]

It is not difficult to verify that (1.3) can be obtained from (5.8).

6

A factorization of $\tau_{G}(W)$

If a simple graph $G=(V,E)$ contains a clique $U$ and a partition $S_{1},S_{2}$ of $V-U$ with $S_{1}\cap N_{G}(S_{2})=\emptyset$, then the chromatic polynomial $\chi(G,\lambda)$ has the following factorization due to Zykov [19] (see [5, 14] also):

[TABLE]

In this section, we find a similar expression for $\tau_{G}(W)$ for any $W\subseteq E(G)-E(G[U])$ by applying results in Sections 2,  3 and 4.

Theorem 6.1

Let $G=(V,E)$ be a connected and loopless multigraph and $U$ be a clique of $G$. If $S_{1}$ and $S_{2}$ form a partition of $V-U$ with $N_{G}[S_{1}]\cap N_{G}[S_{2}]=\emptyset$, as shown in Figure 9, then, for any $W\subseteq E-E(G[U])$,

[TABLE]

where $W_{i}=W\cap E(G[U\cup S_{i}])$.

Proof.  Let $M=E_{G}(U)$, $R=W\cap M$, $G_{i}=G[U\cup S_{i}]$, $M_{i}=M\cap E(G_{i})$, $R_{i}=R\cap E(G_{i})$ and $N_{i}=(W-R)\cap E(G_{i})$ for $i=1,2$. Thus $W_{i}$ is the disjoint union of $R_{i}$ and $N_{i}$. We will prove this result by the following claims.

Claim 1: (6.2) holds if each component of $G[M]$ is a star with a center in $S_{1}\cup S_{2}$.

Let $u_{i}$ represent the vertex in $G_{i}\bullet U$ after identifying all vertices in $U$ as one vertex. For $T_{i}\in{\cal ST}_{G_{i}\bullet U}(N_{i})$ for $i=1,2$, let $T_{1}\cdot T_{2}$ denote the tree obtained from $T_{1}$ and $T_{2}$ by identifying $u_{1}$ and $u_{2}$ as one vertex. By the definition of $G\bullet U$ and the given condition, we have

[TABLE]

Thus, for $i=1,2$, by Theorem 4.2 with $k=1$,

[TABLE]

and

[TABLE]

Thus Claim 1  holds.

Claim 2: (6.2) holds if $R=M$ and $G[M]$ is a forest.

Let $G^{\prime}=G\star W-M$ and $W^{\prime}=E(G\star W)-E(G)$. For $i=1,2$, let $W_{i}=W\cap E(G_{i})$, $G_{i}^{\prime}=G_{i}\star W_{i}-R_{i}$ and $W_{i}^{\prime}=E(G\star W_{i})-E(G_{i})$. By Lemma 2.3 (i),

[TABLE]

and

[TABLE]

Note that $U$ is a clique of $G^{\prime}$ and $G^{\prime}[E_{G^{\prime}}(U)]$ is a star. By Claim 1,

[TABLE]

Thus, Claim 2 follows from (6.5), (6.6) and (6.7).

Claim 3: (6.2) holds.

For any $M^{\prime}\subseteq M$, $G[M^{\prime}]$ is not a forest if and only if $G_{i}[M^{\prime}_{i}]$ is not a forest for some $i=1,2$, where $M^{\prime}_{i}=M^{\prime}\cap E_{G}(U)$. Thus

[TABLE]

Let $R$ be a fixed subset of $M$ such that $G[R]$ is a forest. For $i=1,2$, let ${\cal R}_{i}=\{R^{\prime}_{i}:R_{i}\subseteq R_{i}^{\prime}\subseteq M\cap E(G_{i}),G[R_{i}^{\prime}]\mbox{ is a forest}\}$, where $R_{i}=R\cap E(G_{i})$. Note that

[TABLE]

where for distinct order pairs $(R^{\prime}_{1},R^{\prime}_{2})$ and $(R^{\prime\prime}_{1},R^{\prime\prime}_{2})$ in the above union, the two corresponding sets ${\cal ST}_{G-(M-(R_{1}^{\prime}\cup R^{\prime}_{2}))}(R_{1}^{\prime}\cup R_{2}^{\prime}\cup N_{1}\cup N_{2})$ and ${\cal ST}_{G-(M-(R_{1}^{\prime\prime}\cup R^{\prime\prime}_{2}))}(R_{1}^{\prime\prime}\cup R_{2}^{\prime\prime}\cup N_{1}\cup N_{2})$ are disjoint. Similarly,

[TABLE]

where the above union is disjoint union for both $i=1,2$. By Claim 2, for any $R^{\prime}_{i}\in{\cal R}_{i}$ for $i=1,2$, we have

[TABLE]

Thus, Claim 3 follows from (6.9), (6.10) and (6.11), and the result is proven. $\Box$

Corollary 6.1

Let $G=(V,E)$ be any connected multigraph and $U$ be a clique of $G$. If $w$ is a vertex in $V-U$ with $N_{G}[w]\cap N_{G}[V-(U\cup\{w\})]=\emptyset$, then

[TABLE]

Proof.  Let $S_{1}=\{w\}$ and $S_{2}=V-\{w\}$. As $N_{G}[w]\cap N_{G}[V-(U\cup\{w\})]=\emptyset$, by applying Theorem 6.1,

[TABLE]

where $G_{1}=G[U\cup\{w\}]$ and $G_{2}=G[U\cup S_{2}]=G-w$. By Theorem 4.2,

[TABLE]

The result then follows from (6.13) and (6.14). $\Box$

Remarks: (a) The condition for (6.1) is weaker than the one for (6.2), as (6.2) holds with an extra condition $N_{G}[S_{1}]\cap N_{G}[N_{2}]=\emptyset$.

(b) Note that when $W=\emptyset$, (6.2) is equivalent to the following equality

[TABLE]

when $(x,y)=(1,1)$, where $T_{G}(x,y)$ is the Tutte polynomial of $G$.

(c) Under the condition that $S_{1}\cap N_{G}(S_{2})=\emptyset$, (6.1) implies that (6.15) holds when $y=0$, as $T_{G}(1-x,0)=x^{-c(G)}(-1)^{|V|-c(G)}\chi(G,x)$ holds for any simple graph $G$ (see [3, 7]), where $c(G)$ is the number of components of $G$. Furthermore, (6.15) also holds for graph $G$ satisfying condition $N_{G}[S_{1}]\cap N_{G}[N_{2}]=\emptyset$ when $(x,y)=(2,2)$, as $T_{G}(2,2)=2^{|E(G)|}$ holds for any graph $G$. We have verified (6.15) for some graphs $G$ satisfying the same condition when $(x,y)=(0,-1)$, but we are not sure if it holds for all graphs $G$ satisfying this condition.

Problem 6.1

Let $U$ be a clique of $G=(V,E)$. If $V-U$ has a partition $S_{1}$ and $S_{2}$ with $N_{G}[S_{1}]\cap N_{G}[N_{2}]=\emptyset$, does (6.15) hold at $(x,y)=(0,-1)$?