Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs

TL;DR

This paper investigates conditions under which compositions and products of digraphs can be decomposed into two strong, arc-disjoint subdigraphs, providing new characterizations and sufficient conditions for such decompositions.

Contribution

It offers new sufficient conditions and characterizations for good decompositions in digraph compositions and products, especially when the base digraphs are strong and semicomplete.

Findings

Strong product of two digraphs always has a good decomposition.

Cartesian powers of strong digraphs with cycle covers have good decompositions.

Characterizations for compositions with strong semicomplete digraphs.

Abstract

A digraph $D=(V,A)$ has a good decomposition if $A$ has two disjoint sets $A_1$ and $A_2$ such that both $(V,A_1)$ and $(V,A_2)$ are strong. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ For digraph compositions $Q=T[H_1,\dots H_t]$, we obtain sufficient conditions for $Q$ to have a good decomposition and a characterization of $Q$ with a good decomposition when $T$ is a strong semicomplete digraph and each $H_i$ is an arbitrary digraph with at least two vertices. For digraph products, we prove the following: (a) if $k\geq 2$ is an integer and $G$ is a strong digraph which has a collection of arc-disjoint cycles covering all vertices, then the Cartesian product digraph $G^{\square k}$ (the $k$th powers with respect to Cartesian product) has a good decomposition; (b) for any strong digraphs $G, H$, the strong product $G\boxtimes H$ has a good decomposition.

Equations

$$A(Q)={\cup }_{i=1}^{t}A(H_i)\cup \{u_{i{j}_i}{u}_{p{q}_p}\mid u_i{u}_p\in A(T),1\le j_i\le n_i,1\le q_p\le n_p\}.$$

$$u_1{u}_2,u_2{u}_1,u_3{u}_4,u_4{u}_3,u_1{u}_4,u_2{u}_3,u_4{u}_2,u_3{u}_1.$$

$$\{u_{1,1}{u}_{2,1},u_{2,1}{u}_{1,2},u_{1,2}{u}_{4,1},u_{4,1}{u}_{3,1},u_{3,1}{u}_{1,1}\}$$

$$\{u_{2,1}{u}_{1,1},u_{1,1}{u}_{4,1},u_{4,1}{u}_{2,1},u_{2,1}{u}_{3,1},u_{3,1}{u}_{1,2},u_{1,2}{u}_{2,1}\}.$$

$$\{u_{i,j}{u}_{i+1,j}\mid 1\le i\le t-1,1\le j\le 2\}\cup \{u_{t,1}{u}_{1,2},u_{t,2}{u}_{1,1}\}$$

$$\{u_{i,j}{u}_{i+1,(j+1\text{mbox}mod2)}\mid 1\le i\le t-1,1\le j\le 2\}\cup \{u_{t,1}{u}_{1,1},u_{t,2}{u}_{1,2}\}.$$

$$\{u_{1,1}{u}_{2,1},u_{3,1}{u}_{1,1},u_{1,2}{u}_{2,2},u_{1,2}{u}_{2,3},u_{3,2}{u}_{1,2},u_{3,3}{u}_{1,2},u_{2,1}{u}_{3,2},u_{2,2}{u}_{3,1},u_{2,3}{u}_{3,3}\},$$

$$\{u_{1,1}{u}_{2,2},u_{1,1}{u}_{2,3},u_{3,2}{u}_{1,1},u_{3,3}{u}_{1,1},u_{1,2}{u}_{2,1},u_{3,1}{u}_{1,2},u_{2,1}{u}_{3,3},u_{2,2}{u}_{3,2},u_{2,3}{u}_{3,1}\},$$

$$V(G_h\square G_h)\cap V(P_{h+1}\square P_{h+1})\supseteq \{u_{t,t}\}\text{mbox}andA(G_h\square G_h)\cap A(P_{h+1}\square P_{h+1})=\varnothing .$$

Full text

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For digraph compositions $Q=T[H_{1},\dots H_{t}]$, we obtain sufficient conditions for $Q$ to have a good decomposition and a characterization of $Q$ with a good decomposition when $T$ is a strong semicomplete digraph and each $H_{i}$ is an arbitrary digraph with at least two vertices.

For digraph products, we prove the following: (a) if $k\geq 2$ is an integer and $G$ is a strong digraph which has a collection of arc-disjoint cycles covering all vertices, then the Cartesian product digraph $G^{\square k}$ (the $k$th powers with respect to Cartesian product) has a good decomposition; (b) for any strong digraphs $G,H$, the strong product $G\boxtimes H$ has a good decomposition.

Keywords: strong spanning subdigraph; decomposition into strong spanning subdigraphs; semicomplete digraph; digraph composition; Cartesian product; strong product.

AMS subject classification (2010): 05C20, 05C70, 05C76, 05C85.

1 Introduction

We refer the readers to [1, 2, 6] for graph theoretical notation and terminology not given here. A digraph $D=(V,A)$ is strongly connected (or strong) if there exists a path from $x$ to $y$ and a path from $y$ to $x$ in $D$ for every pair of distinct vertices $x,y$ of $D$. A digraph $D$ is $k$-arc-strong if $D-X$ is strong for every subset $X\subseteq A$ of size at most $k-1$.

An out-branching $B^{+}_{s}$ (respectively, in-branching $B^{-}_{s}$) in a digraph $D=(V,A)$ is a connected spanning subdigraph of $D$ in which each vertex $x\neq s$ has precisely one arc entering (leaving) it and $s$ has no arcs entering (leaving) it. The vertex $s$ is the root of $B^{+}_{s}$ (respectively, $B^{-}_{s}$). Edmonds [9] characterized digraphs with have $k$ arc-disjoint out-branchings rooted at a specified vertex $s.$ Furthermore, there exists a polynomial algorithm for finding $k$ arc-disjoint out-branchings from a given root $s$ if they exist (see p. 346 of [1]). However, if we ask for the existence of a pair of arc-disjoint branchings $B^{+}_{s}$, $B^{-}_{s}$ such that the first is an out-branching rooted at $s$ and the latter is an in-branching rooted at $s$, then the problem becomes NP-complete (see Section 9.6 of [1]). In connection with this problem, Thomassen [12] posed the following conjecture: There exists an integer $N$ so that every $N$-arc-strong digraph $D$ contains a pair of arc-disjoint in- and out-branchings.

Bang-Jensen and Yeo generalized the above conjecture as follows.333Every strong digraph has an out- and in-branching. A digraph $D=(V,A)$ has a good decomposition if $A$ has two disjoint sets $A_{1}$ and $A_{2}$ such that both $(V,A_{1})$ and $(V,A_{2})$ are strong [4].

Conjecture 1.1

[5]** There exists an integer $N$ so that every $N$-arc-strong digraph $D$ contains a pair of arc-disjoint strong spanning subdigraphs.

For a general digraph $D$, it is a hard problem to decide whether $D$ has a decomposition into two strong spanning subdigraphs.

Theorem 1.1

[5]** It is NP-complete to decide whether a digraph contains a pair of arc-disjoint strong spanning subdigraphs.

Clearly, every digraph with a good decomposition is 2-arc-strong. Bang-Jensen and Yeo characterized the semicomplete digraphs with a good decomposition.

Theorem 1.2

[5]** A 2-arc-strong semicomplete digraph $D$ has a pair of arc-disjoint strong spanning subdigraphs if and only if $D$ is not isomorphic to $S_{4}$, where $S_{4}$ is obtained from the complete digraph with four vertices by deleting a cycle of length four. Furthermore, a good decomposition of $D$ can be obtained in polynomial time when it exists.

The following result extends Theorem 1.2 to locally semicomplete digraphs.

Theorem 1.3

[4]** A 2-arc-strong locally semicomplete digraph $D$ has a pair of arc-disjoint strong spanning subdigraphs if and only if $D$ is not the second power of an even cycle.

Let $T$ be a digraph with $t$ vertices $u_{1},\dots,u_{t}$ and let $H_{1},\dots H_{t}$ be digraphs such that $H_{i}$ has vertices $u_{i,j_{i}},\ 1\leq j_{i}\leq n_{i}.$ Then the composition $Q=T[H_{1},\dots,H_{t}]$ is a digraph with vertex set $\{u_{i,j_{i}}\mid 1\leq i\leq t,1\leq j_{i}\leq n_{i}\}$ and arc set

[TABLE]

In this paper, we continue research on good decompositions in classes of digraphs and consider digraph compositions and products.

In Section 2, for digraph compositions $Q=T[H_{1},\dots H_{t}]$, we obtain sufficient conditions for $Q$ to have a good decomposition (Theorem 2.2) and a characterization of $Q$ with a good decomposition when $T$ is a strong semicomplete digraph and each $H_{i}$ is an arbitrary digraph with at least two vertices (Theorem 2.3). Remarkably, in Theorem 2.3 as in Theorem 1.2, there are only a finite number of exceptional digraphs, which for Theorem 2.3 is three. Thus, as Theorems 1.2 and 1.3, Theorem 2.3 confirms Conjecture 1.1 for a special class of digraphs.

In Section 3, for digraph products, we prove the following: (a) if $k\geq 2$ is an integer and $G$ is a strong digraph which arcs can be partitioned into cycles, then the Cartesian product digraph $G^{\square k}$ (the $k$th powers with respect to Cartesian product) has a good decomposition (Theorem 3.4); (b) for any strong digraphs $G,H$, the strong product $G\boxtimes H$ has a good decomposition (Theorem 3.7). Necessary definitions of the digraph products are given in Section 3.

Simple examinations of our constructive proofs show that all our decompositions can be found in polynomial time.

We conclude the paper in Section 4, where we pose a number of open problems.

2 Compositions of digraphs

Let $H^{\prime}_{i}$ denote $H_{i}$ with all arcs deleted, where $1\leq i\leq t$ and let $Q^{\prime}=T[H^{\prime}_{1},\dots,H^{\prime}_{t}].$

Compositions of digraphs is a useful concept in digraph theory, see e.g. [1]. In particular, they are used in the Bang-Jensen-Huang characterization of quasi-transitive digraphs and its structural and algorithmic applications for quasi-transitive digraphs and their extensions, see e.g. [1, 2, 8].

Let us start from a simple observation, which will be useful in the proofs of the theorems of this section.

Lemma 2.1

Let $Q=D[H_{1},\dots,H_{t}].$ If an induced subdigraph $Q^{*}$ of $Q^{\prime}=D[H^{\prime}_{1},\dots,H^{\prime}_{t}]$ with at least one vertex in each $H_{i}$ has a good decomposition, then so have $Q^{\prime}$ and $Q.$

Proof: For every $1\leq i\leq t,$ let $H_{i}^{(m_{i})}$ be the subdigraph of $H^{\prime}_{i}$ induced by $\{u_{i,1},u_{i,2},\dots,u_{i,m_{i}}\},$ where $1\leq m_{i}\leq n_{i}.$ Without loss of generality, let $Q^{*}=D[H_{1}^{(m_{1})},\dots,H^{(m_{t})}_{t}]$ and let $Q^{*}$ have a decomposition into arc-disjoint strong spanning subdigraphs $D_{1},D_{2}.$ To extend this decomposition to $Q^{\prime},$ for every $1\leq i\leq t$ and $j=1,2$, add to $D_{j}$ the vertices $u_{i,m_{i}+1},\dots,u_{i,n_{i}}$ and let them have the same in- and out-neighbors as $u_{i,1}.$ $\Box$

The following theorem gives sufficient conditions for a digraph composition to have a good decomposition. As in Theorem 1.2, $S_{4}$ will denote the digraph obtained from the complete digraph of order 4 by deleting a cycle of length 4.

Theorem 2.2

Let $Q=T[H_{1},\dots,H_{t}],$ where $t\geq 2.$ Then $Q$ has a good decomposition if at least one of the following conditions holds:

(a) $T$ is a 2-arc-strong semicomplete digraph and $H_{1},\dots,H_{t}$ are arbitrary digraphs, but $Q$ is not isomorphic to $S_{4};$

(b) $T$ has a Hamiltonian cycle and either $t$ is even and $n_{i}\geq 2$ for every $i=1,\dots,t$ or $t$ is odd and $n_{i}\geq 3$ for every $i=1,\dots,t$ apart from one $i$ for which $n_{i}\geq 2,$ or $t$ is odd, $n_{i}\geq 2$ for every $i=1,\dots,t$ and at least two distinct subdigraphs $H_{i}$ have arcs.

(c) If $T$ and all $H_{i}$ are strong digraphs of orders at least 2.

Proof: Part (a) If $T$ is not isomorphic to $S_{4}$ then we are done by Theorem 1.2 and Lemma 2.1. Now assume that $T$ is isomorphic to $S_{4}$, but $Q$ is not isomorphic to $S_{4}$. Let the vertices of $T$ be $u_{1},u_{2},u_{3},u_{4}$ and its arcs

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Since $Q$ is not isomorphic to $S_{4}$, at least one of $H_{1},H_{2},H_{3},H_{4}$ has at least two vertices. Without loss of generality, let $H_{1}$ have at least two vertices. Consider the subdigraph $Q^{*}$ of $Q^{\prime}$ induced by $\{u_{1,1},u_{1,2},u_{2,1},u_{3,1},u_{4,1}\}.$ Then $Q^{*}$ has two arc-disjoint strong spanning subdigraphs: $D_{1}$ with arcs

[TABLE]

and $D_{2}$ with arcs

[TABLE]

It remains to apply Lemma 2.1 to obtain a good decomposition of $Q.$

Part (b) Without loss of generality, assume that $u_{1}u_{2}\dots u_{t}u_{1}$ is a Hamiltonian cycle of $T.$ Let $U=\cup_{i=1}^{t}\{u_{i,1},u_{i,2}\}.$

Case 1: $t$ is even and $n_{i}\geq 2$ for every $i=1,\dots,t.$

The following arc sets induce arc-disjoint strong spanning subdigraphs $D_{1},D_{2}$ of $Q^{\prime}[U]:$

[TABLE]

[TABLE]

It remains to apply Lemma 2.1.

Case 2: $t$ is odd, $n_{i}\geq 2$ for every $i=1,\dots,t$ and at least two distinct subdigraphs $H_{i}$ have arcs.

Let $e_{p},e_{q}$ be arcs in two distinct subdigraphs $H_{p}$ and $H_{q}$. We may assume that both end-vertices of $e_{p}$ and $e_{q}$ are in $U.$ Observe that while $D_{1}$ (with arcs listed in (1)) is strong, $D_{2}$ (with arcs listed in (2)) forms two arc-disjoint cycles $C$ and $Z.$ We may assume that the tail (head) of $e_{p}$ ($e_{q}$) is in $C$ and and the head (tail) of $e_{p}$ ($e_{q}$) is in $Z$ (otherwise, relabel vertices in $\{u_{p,1},u_{p,2}\}$ and/or $\{u_{q,1}u_{q_{2}}\}$). Thus, adding $e_{p}$ and $e_{q}$ to $D_{2}$ makes it strong. To obtain two arc-disjoint strong spanning subdigraphs of $Q$ from $D_{1},D_{2}$, let every vertex $u_{i,j}$ for $j\geq 3$ and $1\leq i\leq t$ have the same out- and in-neighbors as $u_{i,1}$ in $Q^{\prime}.$

Case 3: $t$ is odd and $n_{i}\geq 3$ for every $i=1,\dots,t$ apart from one $i$ for which $n_{i}\geq 2.$

Without loss of generality, assume that $n_{1}\geq 2$ and $n_{i}\geq 3$ for all $2\leq i\leq t.$

First we consider the subcase in which $t=3$, $n_{1}=2,$ and $n_{2}=n_{3}=3.$ Then $Q^{\prime}$ has two arc-disjoint spanning subdigraphs $D_{1}$ and $D_{2}$ with arc sets

[TABLE]

[TABLE]

respectively. It is not hard to see that $D_{1}$ and $D_{2}$ are strong by constructing closed walks through all vertices.

Now we extend the previous subcase to that in which $n_{1}=2$ and $n_{i}=3$ for all $2\leq i\leq t.$ First replace index 3 in every vertex of the form $u_{3,i}$ by $t$ in the two arc sets of the previous subcase. Then replace every arc of the form $u_{2,i}u_{t,j}$ in $D_{1}$ by the path $u_{2,i}u_{3,i}\dots u_{t-1,i}u_{t,j}.$ In $D_{2}$, we replace $u_{2,1}u_{t,3}$ by the path $u_{2,1}u_{3,2}u_{4,1}u_{5,2}\dots u_{t-1,1}u_{t,3}$, replace $u_{2,2}u_{t,2}$ by the path $u_{2,2}u_{3,1}u_{4,2}u_{5,1}\dots u_{t-1,2}u_{t,2}$, replace $u_{2,3}u_{t,1}$ by the path $u_{2,3}u_{3,2}u_{4,3}u_{5,2}\dots u_{t-1,3}u_{t,1}$, and finally add the path $u_{2,2}u_{3,3}u_{4,2}u_{5,3}\dots u_{t-1,2}$.

Finally, we extend the previous subcase to the general one using Lemma 2.1.

Part (c) For $j=1,2$, let $T_{j}$ be the subdigraph of $Q$ induced by vertex set $\{u_{i,j}\mid 1\leq i\leq t\}$. Clearly, $T_{1}\cong T_{2}\cong T$ and $T_{1}$ and $T_{2}$ are strong.

Let $Q_{1}$ be the spanning subdigraph of $Q$ with arc set $A(Q_{1})=A(T_{1})\cup(\bigcup_{i=1}^{t}{A(H_{i})})$. Observe that $Q_{1}$ is strong since $T_{1}$ and each $H_{i}$ are strong, and $T_{1}$ has a common vertex with each $H_{i}$, where $1\leq i\leq t$.

Let $Q_{2}$ be the spanning subdigraph of $Q$ with arc set $A(Q_{2})=A(Q)\setminus A(Q_{1})$. To see that $Q_{2}$ is strong, we only need to find a strong subdigraph in $Q_{2}$ which contains $x$ and $y$ for each pair of distinct vertices $x$ and $y$ in $Q_{2}$. We will consider two cases.

Case 1: $x\in V(T_{1})$.

Without loss of generality, we assume that $x=u_{1,1}$ and $y\in\{u_{1,2},u_{2,1},u_{2,2}\}$. We first consider the subcase that $y=u_{2,1}$. Observe that there is at least one arc entering and one arc leaving $u_{1,2}~{}(u_{2,2})$ in $T_{2}$, and so there are two arcs, say $a$ and $b$ ($c$ and $d$), with opposite directions between $x~{}(y)$ and $T_{2}$ in $Q_{2}$. Then by adding the arcs $a,b,c,d$, and the vertices $x,y$ to $T_{2}$, we obtain a strong subdigraph $T_{2}^{\prime}$ of $Q_{2}$ which contains both $x$ and $y$, as desired. For the case that $y\in\{u_{1,2},u_{2,2}\}$, we just add the arcs $a,b$, and the vertex $x$ to $T_{2}$, and then obtain a strong subdigraph $T_{2}^{\prime\prime}$ of $Q_{2}$ which contains both $x$ and $y$.

Case 2: $x\not\in V(T_{1})$.

Without loss of generality, we assume that $x=u_{1,2}$ and $y\in\{u_{1,1},u_{2,1},u_{1,3},u_{2,2},u_{2,3}\}$ (if $u_{1,3}$ and $u_{2,3}$ exist). By Case 1 and the fact that $T_{2}\cong T$ is strong, we are done if $y\in\{u_{1,1},u_{2,2}\}$. For the case that $y=u_{2,1}$, by adding the arcs $c,d$ and the vertex $y$ to $T_{2}$, we can obtain a strong subdigraph $T_{2}^{\prime\prime\prime}$ of $Q_{2}$ which contains both $x$ and $y$. With a similar argument, we can get the desired strong subdigraph for the case that $y\in\{u_{1,3},u_{2,3}\}$.

Hence, we complete the argument and conclude that $Q$ has a good decomposition. $\Box$

We will use Theorem 2.2 to prove the following characterization for certain compositions $T[H_{1},\dots,H_{t}]$, where $T$ is a strong semicomplete digraph. In the characterization, $\overline{K_{p}}$ will stand for the digraph of order $p$ with no arcs. Also, $\overrightarrow{C}_{k}$ and $\overrightarrow{P}_{k}$ will denote the cycle and path with $k$ vertices, respectively.

Theorem 2.3

*Let $T$ be a strong semicomplete digraph on $t\geq 2$ vertices and let $H_{1},\dots,H_{t}$ be arbitrary digraphs, each with at least two vertices. Then $Q=T[H_{1},\dots,H_{t}]$ has a good decomposition if and only if $Q$ is not isomorphic to one of the following three digraphs: $\overrightarrow{C}_{3}[\overline{K_{2}},\overline{K_{2}},\overline{K_{2}}]$, $\overrightarrow{C}_{3}[\overrightarrow{P_{2}},\overline{K_{2}},\overline{K_{2}}].$ $\overrightarrow{C}_{3}[\overline{K_{2}},\overline{K_{2}},\overline{K_{3}}].$ *

Proof: Let us first prove the ‘only if’ part of the theorem, i.e. $\overrightarrow{C}_{3}[\overline{K_{2}},\overline{K_{2}},\overline{K_{2}}],$ $\overrightarrow{C}_{3}[\overrightarrow{P}_{2},\overline{K_{2}},\overline{K_{2}}]$ and $\overrightarrow{C}_{3}[\overline{K_{2}},\overline{K_{2}},\overline{K_{3}}]$ do not have good decompositions. By Lemma 2.1, it suffices to show that neither $\overrightarrow{C}_{3}[\overrightarrow{P}_{2},\overline{K_{2}},\overline{K_{2}}]$ nor $\overrightarrow{C}_{3}[\overline{K_{2}},\overline{K_{2}},\overline{K_{3}}]$ has a good decomposition. The proof is by reductio ad absurdum.

Suppose that $Q=\overrightarrow{C}_{3}[\overrightarrow{P_{2}},\overline{K_{2}},\overline{K_{2}}]$ has a decomposition into two strong spanning subdigraphs $Q_{1},Q_{2}.$ Since $Q$ has 13 arcs, without loss of generality, we may assume that $Q_{1}$ is a Hamiltonian cycle of $Q.$ Since the arc of $H_{1}$ cannot be in a Hamiltonian cycle of $Q$, without loss of generality, let $Q_{1}=u_{1,1}u_{2,1}u_{3,1}u_{1,2}u_{2,2}u_{3,2}u_{1,1}$. Then the remaining arcs of $Q$ form two disjoint cycles $u_{1,1}u_{2,2}u_{3,1}u_{1,1}$ and $u_{1,2}u_{2,1}u_{3,2}u_{1,2}$ and a single arc between them, a contradiction to the assumption that $Q_{2}$ is strong.

Suppose that $Q=\overrightarrow{C}_{3}[\overline{K_{2}},\overline{K_{2}},\overline{K_{3}}]$ has a decomposition into two strong spanning subdigraphs $Q_{1},Q_{2}.$ Since $Q$ has 16 arcs and has no Hamiltonian cycle, each of $Q_{1},Q_{2}$ has 8 arcs. Since $Q$ has only cycles of lengths 3 and 6 and $Q_{1}$ is strong, without loss of generality, we may assume that $Q_{1}$ consists of a cycle $u_{1,1}u_{2,1}u_{3,1}u_{1,2}u_{2,2}u_{3,2}u_{1,1}$ and a path $u_{2,1}u_{3,3}u_{1,1}.$ Then $Q_{2}$ consists of two cycles $u_{1,1}u_{2,2}u_{3,1}u_{1,1}$ and $u_{1,2}u_{2,1}u_{3,2}u_{1,2}$ and a path $u_{2,2}u_{3,3}u_{1,2}.$ Observe that $Q_{2}$ is not strong, a contradiction.

Now we will show the ‘if’ part of the theorem by reductio ad absurdum as well. Assume that $Q$ is not isomorphic to either of the three digraphs, but has no good decomposition.

By Camion’s Theorem [7], $T$ has a Hamiltonian cycle $C=u_{1}u_{2}\dots u_{t}u_{1}$. Thus, Conditions (b) of Theorem 2.2 are applicable. By the conditions, $t$ must be odd and for at least two distinct indexes $p,q\in\{1,2,\dots,t\}$, we have $n_{p}=n_{q}=2.$

Suppose $t\geq 5.$ Then there will be arcs between $H_{i}$ and $H_{i+2}$ in $Q$ for every $i=1,2,\dots,t-2.$ Recall Case 2 of Part (b) of the proof of Theorem 2.2. The arcs between $H_{i}$ and $H_{i+2}$ arcs can be used to make $D_{2}$ strong instead of arcs $e_{p}$ and $e_{q}$ used in Case 2 of Part (b) of the proof of Theorem 2.2. Thus, $Q$ has a good decomposition, a contradiction. Hence, $t=3$ and, without loss of generality, $n_{1}=n_{2}=2$ and $n_{3}\geq 2.$

Suppose that $T$ has opposite arcs. One of these arcs will not be on the Hamiltonian cycle $C$ of $T$ and will correspond to four or more arcs in $Q.$ Now recall Case 2 of Part (b) of the proof of Theorem 2.2. Two of the above-mentioned arcs can be used to make $D_{2}$ strong instead of arcs $e_{p}$ and $e_{q}$ used in Case 2 of Part (b) of the proof of Theorem 2.2. Thus, $Q$ has a good decomposition, a contradiction. Hence, $T=\overrightarrow{C}_{3}.$

Suppose that $n_{3}\geq 4.$ To get a contradiction, by Lemma 2.1 it suffices to show that $Q=\overrightarrow{C}_{3}[\overline{K_{2}},\overline{K_{2}},\overline{K_{4}}]$ has a decomposition into two strong spanning subdigraphs $D_{1},D_{2},$ where $D_{1}$ consists of a cycle $u_{1,1}u_{2,1}u_{3,1}u_{1,2}u_{2,2}u_{3,2}u_{1,1}$ and two paths $u_{2,1}u_{3,4}u_{1,1}$ and $u_{2,2}u_{3,3}u_{1,2}$ and $D_{2}$ consists of two cycles $u_{1,1}u_{2,2}u_{3,1}u_{1,1}$ and $u_{1,2}u_{2,1}u_{3,2}u_{1,2}$ and two paths $u_{2,1}u_{3,3}u_{1,1}$ and $u_{2,2}u_{3,4}u_{1,2}.$ Thus, $n_{3}\leq 3.$

Now consider the case of $n_{1}=n_{2}=2$ and $n_{3}=3.$ Since $Q$ is not isomorphic to $\overrightarrow{C}_{3}[\overline{K_{2}},\overline{K_{2}},\overline{K_{3}}],$ it has an arc in either $H_{1}$ or $H_{2}$ or $H_{3}$, and by Conditions (b) of Theorem 2.2, only one of $H_{1},H_{2},H_{3}$ has an arc $a.$ Without loss of generality, assume that if $H_{1}$ has an arc then $a=u_{1,2}u_{1,1}$, if $H_{2}$ has an arc then $a=u_{2,1}u_{2,2}$ and if $H_{3}$ has an arc then $a=u_{3,2}u_{3,1}.$ Then $Q$ has a decomposition into two spanning subdigraphs $D_{1},D_{2},$ where $D_{1}$ consists of a cycle $u_{1,1}u_{2,1}u_{3,1}u_{1,2}u_{2,2}u_{3,2}u_{1,1}$ and a path $u_{2,1}u_{3,3}u_{1,1}$ and $D_{2}$ consists of two cycles $u_{1,1}u_{2,2}u_{3,1}u_{1,1}$ and $u_{1,2}u_{2,1}u_{3,2}u_{1,2}$, a path $u_{2,2}u_{3,3}u_{1,2}$ and arc $a$. Observe that both $D_{1}$ and $D_{2}$ are strong, a contradiction.

It remains to consider the case of $n_{1}=n_{2}=n_{3}=2.$ Since $Q$ is not isomorphic to $\overrightarrow{C}_{3}[\overline{K_{2}},\overline{K_{2}},\overline{K_{2}}]$, at least one of $H_{1},H_{2}$ and $H_{3}$ has an arc. By Conditions (b) of Theorem 2.2, only one of $H_{1},H_{2}$ and $H_{3}$ has an arc. Without loss of generality, assume that $H_{1}$ has an arc. Suppose that $H_{1}$ has two arcs. Then $H_{1}=\overrightarrow{C}_{2}$. Then we can use the arcs of $H_{1}$ to make $D_{2}$ strong instead of arcs $e_{p}$ and $e_{q}$ used in Case 2 of Part (b) of the proof of Theorem 2.2. Thus, $Q$ has a good decomposition, a contradiction. Hence, if $H_{1}$ has an arc, it must have just one arc. This concludes our proof. $\Box$

3 Products of digraphs

The Cartesian product $G\square H$ of two digraphs $G$ and $H$ is a digraph with vertex set $V(G\square H)=V(G)\times V(H)=\{(x,x^{\prime})\mid x\in V(G),x^{\prime}\in V(H)\}$ and arc set $A(G\square H)=\{(x,x^{\prime})(y,y^{\prime})\mid xy\in A(G),x^{\prime}=y^{\prime},~{}or~{}x=y,x^{\prime}y^{\prime}\in A(H)\}.$ By definition, we know the Cartesian product is associative and commutative, and $G\square H$ is strongly connected if and only if both $G$ and $H$ are strongly connected [10]. We define the $n$th powers with respect to Cartesian product as $D^{\square n}=D\square D\square\cdots\square D$.

In the argument of this section, we will use the following terminology and notation. Let $G$ and $H$ be two digraphs with $V(G)=\{u_{i}\mid 1\leq i\leq n\}$ and $V(H)=\{v_{j}\mid 1\leq j\leq m\}$. For simplicity, we let $u_{i,j}=(u_{i},v_{j})$ for $1\leq i\leq n,1\leq j\leq m$. We use $G(v_{j})$ to denote the subdigraph of $G\square H$ induced by vertex set $\{u_{i,j}\mid 1\leq i\leq n\}$ where $1\leq j\leq m$, and use $H(u_{i})$ to denote the subdigraph of $G\square H$ induced by vertex set $\{u_{i,j}\mid 1\leq j\leq m\}$ where $1\leq i\leq n$. Clearly, we have $G(v_{j})\cong G$ and $H(u_{i})\cong H$. (For example, as shown in Figure 1, $G(v_{j})\cong G$ for $1\leq j\leq 4$ and $H(u_{i})\cong H$ for $1\leq i\leq 3$.) For $1\leq j_{1}\neq j_{2}\leq m$, $u_{i,j_{1}}$ and $u_{i,j_{2}}$ belong to the same digraph $H(u_{i})$ where $u_{i}\in V(G)$; we call $u_{i,j_{2}}$ the vertex corresponding to $u_{i,j_{1}}$ in $G(v_{j_{2}})$; for $1\leq i_{1}\neq i_{2}\leq n$, we call $u_{i_{2},j}$ the vertex corresponding to $u_{i_{1},j}$ in $H(u_{i_{2}})$. Similarly, we can define the subdigraph corresponding to some other subdigraph. For example, in Fig. 1(c), let $P_{1}$ $(P_{2})$ be the path labelled 1 (2) in $H(u_{1})~{}(H(u_{2}))$, then $P_{2}$ is called the path corresponding to $P_{1}$ in $H(u_{2})$.

Lemma 3.1

For any integer $n\geq 2$, the product digraph $D=\overrightarrow{C}_{n}\square\overrightarrow{C}_{n}$ can be decomposed into two arc-disjoint Hamiltonian cycles.

Proof: Let $G=H\cong\overrightarrow{C}_{n}$; moreover $G=u_{1}u_{2}\dots u_{n}u_{1}$ and $H=v_{1}v_{2}\dots v_{n}v_{1}.$ Let $P_{i}=G(v_{i})-u_{n-i,i}u_{n+1-i,i}$ for $1\leq i\leq n-1$ and $P_{n}=G(v_{n})-u_{n,n}u_{1,n}$. Let $D^{\prime}$ be the subdigraph of $D$ which is a union of $n$ paths $P_{i}$ and the following $n$ arcs: $\{u_{n-i,i}u_{n-i,i+1}\mid 1\leq i\leq n-1\}\cup\{u_{n,n}u_{n,1}\}.$ Let $D^{\prime\prime}$ be a spanning subdigraph of $D$ with $A(D^{\prime\prime})=A(D)\setminus A(D^{\prime})$. It is not hard to check that both $D^{\prime}$ and $D^{\prime\prime}$ are Hamiltonian cycles of $D$; this completes the proof. $\Box$

Note that deciding whether a digraph $D$ has a collection of arc-disjoint cycle covering all vertices of $D$, can be done in polynomial time using network flows. Indeed, assign lower bound 1 and upper bound $\min\{d^{-}(x),d^{+}(x)\}$ to every vertex $x$ in $D$ and lower bound 0 and upper bound 1 to every arc of $D$. Observe that the resulting network has a feasible flow if and only if $D$ has a collection of arc-disjoint cycle covering all vertices of $D$. Observe that the existence of a flow in a network with lower and upper bounds on vertices and arcs can be decided in polynomial time, see e.g. Chapter 4 in [1]. Moreover, we can compute such a flow in polynomial time (if it exists) and obtain the corresponding collection of cycles in $D.$ The following lemma may be of independent interest.

Lemma 3.2

Let $G$ be a strong digraph of order at least two which has a collection of arc-disjoint cycle covering all its vertices. Then the product digraph $D=G\square G$ can be decomposed into two arc-disjoint strong spanning subdigraphs. Moreover, these two arc-disjoint strong spanning subdigraphs can be found in polynomial time.

Proof: By the arguments in the paragraph before this lemma, we may assume that we are given a collection $(P_{0},P_{1},P_{2},\cdots,P_{p})$ of arc-disjoint cycle covering all vertices of $G$. For each $h=0,1,2,\cdots,p$, let $G_{h}$ denote the digraph with vertices $\bigcup_{i=0}^{h}{V(P_{i})}$ and arcs $\bigcup_{i=0}^{h}{A(P_{i})}$. Now we will prove the lemma by induction on the number of cycles in the collection.

For the base step, by Lemma 3.1, we have that $G_{0}\square G_{0}=P_{0}\square P_{0}$ can be decomposed into two arc-disjoint strong spanning subdigraphs.

For the inductive step, we assume that $G_{h}\square G_{h}~{}(0\leq h\leq p-1)$ can be decomposed into two arc-disjoint strong spanning subdigraphs $D_{h}^{\prime}$ and $D_{h}^{\prime\prime}$. We will construct two arc-disjoint strong spanning subdigraphs in $G_{h+1}\square G_{h+1}$.

If $V(G_{h})\subseteq V(P_{h+1})$, then $P_{h+1}$ is a Hamiltonian cycle of $G_{h+1}$, and we are done by Lemma 3.1. If $V(P_{h+1})\subseteq V(G_{h})$, then $G_{h}$ is a strong spanning subdigraph of $G_{h+1}$, and we are also done.

In the following argument, we assume that $V(G_{h})\setminus V(P_{h+1})\neq\emptyset$ and $V(P_{h+1})\setminus V(G_{h})\neq\emptyset$. Without loss of generality, for the first copies of $G_{h}$ and $P_{h+1}$ in $G_{h}\square G_{h}$ and $P_{h+1}\square P_{h+1}$, let $V(G_{h})=\{u_{i}\mid 1\leq i\leq t\}$, $V(P_{h+1})=\{u_{i}\mid s\leq i\leq\ell\}$. We have $1<s\leq t<\ell$. For the second copies of $G_{h}$ and $P_{h+1}$ in $G_{h}\square G_{h}$ and $P_{h+1}\square P_{h+1}$, we will use $v_{i}$’s rather than $u_{i}$’s.

By Lemma 3.1, in $G_{h+1}\square G_{h+1}$, the subdigraph $P_{h+1}\square P_{h+1}$ can be decomposed into two arc-disjoint strong spanning subdigraphs $\overline{D}_{h}^{\prime}$ and $\overline{D}_{h}^{\prime\prime}$. Observe that

[TABLE]

For $1\leq j\leq s-1$, let $G_{h,j}$ be the subdigraph of $G(v_{j})$ corresponding to $P_{h+1}$. For $t+1\leq j\leq\ell$, let $G_{h,j}$ be the subdigraph of $G(v_{j})$ corresponding to $G_{h}$. For $1\leq i\leq s-1$, let $H_{h,i}$ be the subdigraph of $H(u_{i})$ corresponding to $P_{h+1}$. For $t+1\leq i\leq\ell$, let $H_{h,i}$ be the subdigraph of $H(u_{i})$ corresponding to $G_{h}$.

Now let $D_{h+1}^{\prime}$ be a union of the following strong digraphs: $D_{h}^{\prime}$, $\overline{D}_{h}^{\prime}$, $H_{h,i}$ and $G_{h,j}$ for all $t+1\leq i,j\leq\ell$. Observe that $D_{h+1}^{\prime}$ is a strong spanning subdigraph of $G_{h+1}\square G_{h+1}$ since $\overline{D}_{h}^{\prime}$ has at least one common vertex with each of $D_{h}^{\prime}$, $H_{h,i}$ and $G_{h,j}$ for all $t+1\leq i,j\leq\ell$. Let $D_{h+1}^{\prime\prime}$ be a spanning subdigraph of $G_{h+1}\square G_{h+1}$ with $A(D_{h+1}^{\prime\prime})=A(G_{h+1}\square G_{h+1})\setminus A(D_{h+1}^{\prime})$. Observe that $D_{h+1}^{\prime\prime}$ is the union of $D_{h}^{\prime\prime}$, $\overline{D}_{h}^{\prime\prime}$, $H_{h,i}$ and $G_{h,j}$ for all $1\leq i,j\leq s-1$. And $D_{h}^{\prime\prime}$ has at least one common vertex with each of $\overline{D}_{h}^{\prime\prime}$, $H_{h,i}$ and $G_{h,j}$ for all $1\leq i,j\leq s-1$, thus $D_{h+1}^{\prime\prime}$ is strong.

Hence, we complete the inductive step and conclude that $D=G\square G$ can be decomposed into two arc-disjoint strong spanning subdigraphs. Moreover, by the above argument, these subdigraphs can be found in polynomial time. $\Box$

Lemma 3.3

For any two strong digraphs $G$ and $H$, if $G$ contains a pair of arc-disjoint strong spanning subdigraphs, then the product digraph $D=G\square H$ can be decomposed into two arc-disjoint strong spanning subdigraphs.

Proof: Let $V(G)=\{u_{i}\mid 1\leq i\leq n\},V(H)=\{v_{j}\mid 1\leq j\leq m\}$, and $G$ contain two arc-disjoint strong spanning subdigraphs $G_{1}$ and $G_{2}$. For $1\leq j\leq m$, let $G_{1,j}$ be the subdigraph of $G(v_{j})$ corresponding to $G_{1}$. Let $D^{\prime}$ be a union of $H(u_{1})$ and $G_{1,j}$ for all $1\leq j\leq m$, and $D^{\prime\prime}$ be a subdigraph of $D$ with $V(D^{\prime\prime})=V(D)$ and $A(D^{\prime\prime})=A(D)\setminus A(D^{\prime})$. It is not hard to verify that both $D^{\prime}$ and $D^{\prime\prime}$ are strong spanning subdigraphs of $D$. This completes the proof. $\Box$

By the definition of $D^{\square k}$, associativity of the Cartesian product, and Lemmas 3.2 and 3.3, we can obtain the following result on $G^{\square k}$ for any integer $k\geq 2$.

Theorem 3.4

Let $G$ be a strong digraph of order at least two which has a collection of arc-disjoint cycle covering all its vertices and let $k\geq 2$ be an integer. Then the product digraph $D=G^{\square k}$ can be decomposed into two arc-disjoint strong spanning subdigraphs. Moreover, for any fixed integer $k$, these two subdigraphs can be found in polynomial time.

The strong product $G\boxtimes H$ of two digraphs $G$ and $H$ is a digraph with vertex set $V(G\boxtimes H)=V(G)\times V(H)=\{(x,x^{\prime})\mid x\in V(G),x^{\prime}\in V(H)\}$ and arc set $A(G\boxtimes H)=\{(x,x^{\prime})(y,y^{\prime})\mid xy\in A(G),x^{\prime}=y^{\prime},~{}or~{}x=y,x^{\prime}y^{\prime}\in A(H),~{}or~{}xy\in A(G),~{}x^{\prime}y^{\prime}\in A(H)\}.$ By definition, $G\square H$ is a spanning subdigraph of $G\boxtimes H$, and $G\boxtimes H$ is strongly connected if and only if both $G$ and $H$ are strongly connected [10]. In the following argument, we will still use the terminology and notation introduced earlier in this section, since $G\square H$ is a spanning subdigraph of $G\boxtimes H$.

Lemma 3.5

For any two integers $n,m\geq 2$, the product digraph $D=\overrightarrow{C}_{n}\boxtimes\overrightarrow{C}_{m}$ can be decomposed into two arc-disjoint strong spanning subdigraphs.

Proof: Let $\overrightarrow{C}_{n}=u_{1}u_{2}\dots u_{n}u_{1}$ and $\overrightarrow{C}_{m}=v_{1}v_{2}\dots v_{m}.$ Let $D^{\prime}$ be the spanning subdigraph of $D$ which is the union of $G(v_{j})$ for $1\leq j\leq m$ and the following additional $m$ arcs: $\{u_{n,j}u_{1,j+1}\mid 1\leq j\leq m-1\}\cup\{u_{1,m}u_{2,1}\}$. Observe that $D^{\prime}$ is strong. Let $D^{\prime\prime}$ be a spanning subdigraph of $D$ with $A(D^{\prime\prime})=A(D)\setminus A(D^{\prime})$. To see that $D^{\prime\prime}$ is strong, observe that it contains $H(u_{i})$ for $1\leq i\leq n$ and arcs $\{u_{i,1}u_{i+1,2}\mid 1\leq i\leq n-1\}\cup\{u_{n,m}u_{1,1}\}.$ $\Box$

We will use the following decomposition of strong digraphs.

An ear decomposition of a digraph $D$ is a sequence $\mathcal{P}=(P_{0},P_{1},P_{2},\cdots,P_{t})$, where $P_{0}$ is a cycle or a vertex and each $P_{i}$ is a path, or a cycle with the following properties:

$(a)$ $P_{i}$ and $P_{j}$ are arc-disjoint when $i\neq j$.

$(b)$ For each $i=0,1,2,\cdots,t$: let $D_{i}$ denote the digraph with vertices $\bigcup_{j=0}^{i}{V(P_{j})}$ and arcs $\bigcup_{j=0}^{i}{A(P_{j})}$. If $P_{i}$ is a cycle, then it has precisely one vertex in common with $V(D_{i-1})$. Otherwise the end vertices of $P_{i}$ are distinct vertices of $V(D_{i-1})$ and no other vertex of $P_{i}$ belongs to $V(D_{i-1})$.

$(c)$ $\bigcup_{j=0}^{t}{A(P_{j})}=A(D)$.

The following result is well-known, see e.g. [1].

Theorem 3.6

Let $D$ be a digraph with at least two vertices. Then $D$ is strong if and only if it has an ear decomposition. Furthermore, if $D$ is strong, every cycle can be used as a starting cycle $P_{0}$ for an ear decomposition of $D$, and there is a linear-time algorithm to find such an ear decomposition.

Theorem 3.7

For any strong digraphs $G$ and $H$ with orders at least 2, the product digraph $D=G\boxtimes H$ can be decomposed into two arc-disjoint strong spanning subdigraphs. Moreover, these two arc-disjoint strong spanning subdigraphs can be found in polynomial time.

Proof: By Theorem 3.7 $G$ has an ear decomposition $\mathcal{P}=(P_{0},P_{1},P_{2},\cdots,P_{p})$ and $H$ has an ear decomposition $\mathcal{Q}=(Q_{0},Q_{1},Q_{2},\cdots,Q_{q})$, such that $P_{0}$ is a cycle of $G$ and $Q_{0}$ is a cycle of $H$ by Theorem 3.6. Let $G_{i}$ denote the subdigraph of $G$ with vertices $\bigcup_{j=0}^{i}{V(P_{j})}$ and arcs $\bigcup_{j=0}^{i}{A(P_{j})}$ and let $H_{i}$ denote the subdigraph of $H$ with vertices $\bigcup_{j=0}^{i}{V(Q_{j})}$ and arcs $\bigcup_{j=0}^{i}{A(Q_{j})}.$

We will prove the theorem by induction on $r\in\{0,1,\dots,p+q\}.$ For the base step, by Lemma 3.5, we have that $P_{0}\boxtimes Q_{0}$ can be decomposed into two arc-disjoint strong spanning subdigraphs. For the inductive step, we assume that $r=h+g<p+q\ (h\leq p,g\leq q)$ and $G_{h}\boxtimes H_{g}$ can be decomposed into two arc-disjoint strong spanning subdigraphs $D^{\prime}$ and $D^{\prime\prime}$.

Since strong product is a commutative operation, without loss of generality it suffices to prove that $G_{h+1}\boxtimes H_{g}$ $(h<p)$ can be decomposed into two arc-disjoint strong spanning subdigraphs. Let $V(G_{h})=\{u_{1},u_{2},\dots,u_{\ell}\}$, $V(H_{g})=\{v_{1},v_{2},\dots,v_{m}\}$ and $v_{1}v_{s}\in A(H_{g}).$ Let $P_{h+1,j}$ be the subdigraph of $G(v_{j})$ corresponding to $P_{h+1}$ for $1\leq j\leq m.$ We will consider two cases.

Case 1: $P_{h+1}$ is a cycle.

Let $P_{h+1}=u_{\ell}u_{\ell+1}\dots u_{n}u_{\ell}.$ Observe that every $P_{h+1,j}$ for $1\leq j\leq m$ shares vertex $u_{\ell,j}$ with $D^{\prime}.$ Thus, the union $U_{1}$ of $D^{\prime}$ and $P_{h+1,j}$ for $1\leq j\leq m$ is a strong spanning subdigraph of $G_{h+1}\boxtimes H_{g}.$ Let $V(U_{2})=V(G_{h+1}\boxtimes H_{g})$ and $A(U_{2})=A(G_{h+1}\boxtimes H_{g})\setminus A(U_{1}).$

Observe that $A(U_{2})$ contains $A(D^{\prime\prime})$, $A(H(u_{i}))$ for $\ell+1\leq i\leq n$ and $\{u_{i,1}u_{i+1,s}\mid\ell\leq i\leq n-1\}\cup\{u_{n,1}u_{\ell,s}\}.$ Thus, $U_{2}$ is strong.

Case 2: $P_{h+1}$ is a path.

Let $P_{h+1}=u_{\ell}u_{\ell+1}\dots u_{n-1}u_{t},$ where $t<\ell.$ Let $U_{1}$ be the union of $D^{\prime}$ and $P_{h+1,j}$ for $1\leq j\leq m.$ Observe that $U_{1}$ is a spanning subdigraph of $G_{h+1}\boxtimes H_{g}$ and strong since every $P_{h+1,j}$ for $1\leq j\leq m$ shares its end-vertices with $D^{\prime}.$ Let $V(U_{2})=V(G_{h+1}\boxtimes H_{g})$ and $A(U_{2})=A(G_{h+1}\boxtimes H_{g})\setminus A(U_{1}).$ Observe that $A(U_{2})$ contains $A(D^{\prime\prime})$, $A(H(u_{i}))$ for $\ell+1\leq i\leq n-1$ and $\{u_{i,1}u_{i+1,s}\mid\ell\leq i\leq n-2\}\cup\{u_{n-1,1}u_{t,s}\}.$ Thus, $U_{2}$ is strong.

Hence, we complete the inductive step and conclude that $D=G\boxtimes H$ can be decomposed into two arc-disjoint strong spanning subdigraphs. Furthermore, by Theorem 3.6, the proof of Lemma 3.5, and the argument of this theorem, we can conclude that these two strong spanning subdigraphs can be found in polynomial time. $\Box$

The lexicographic product $G\circ H$ of two digraphs $G$ and $H$ is a digraph with vertex set $V(G\circ H)=V(G)\times V(H)=\{(x,x^{\prime})\mid x\in V(G),x^{\prime}\in V(H)\}$ and arc set $A(G\circ H)=\{(x,x^{\prime})(y,y^{\prime})\mid xy\in A(G),~{}or~{}x=y~{}and~{}x^{\prime}y^{\prime}\in A(H)\}$[10]. By definition, $G\boxtimes H$ is a spanning subdigraph of $G\circ H$, so the following result holds by Theorem 3.7: For any strong connected digraphs $G$ and $H$ with orders at least 2, the product digraph $D=G\circ H$ can be decomposed into two arc-disjoint strong spanning subdigraphs. Moreover, these two arc-disjoint strong spanning subdigraphs can be found in polynomial time. In fact, we can get a more general result.

A digraph is Hamiltonian decomposable if it has a family of Hamiltonian dicycles such that every arc of the digraph belongs to exactly one of the dicycles. Ng [11] gives the most complete result among digraph products.

Theorem 3.8

[11]** If $G$ and $H$ are Hamiltonian decomposable digraphs, and $|V(G)|$ is odd, then $G\circ H$ is Hamiltonian decomposable.

Theorem 3.8 implies that if $G$ and $H$ are Hamiltonian decomposable digraphs, and $|V(G)|$ is odd, then $G\circ H$ can be decomposed into two arc-disjoint strong spanning subdigraphs. It is not hard to extend this result as follows: for any strong digraphs $G$ and $H$ of orders at least 2, if $H$ contains $\ell\geq 1$ arc-disjoint strong spanning subdigraphs, then the product digraph $D=G\circ H$ can be decomposed into $\ell+1$ arc-disjoint strong spanning subdigraphs.

4 Open Problems

We have characterized digraphs $T[H_{1},\dots,H_{t}],$ where $T$ is strong semicomplete and every $H_{i}$ is arbitrary with at least two vertices, which have a good decomposition. It is a natural open problem to extend the characterization to all such digraphs, where some $H_{i}$’s can have just one vertex. Of course, the extended characterization would generalize also Theorem 1.2.

A digraph $Q$ is quasi-transitive, if for any triple $x,y,z$ of distinct vertices of $Q$, if $xy$ and $yz$ are arcs of $Q$ then either $xz$ or $zx$ or both are arcs of $Q.$ For a recent survey on quasi-transitive digraphs and their generalizations, see a chapter [8] by Galeana-Sánchez and Hernández-Cruz. Bang-Jensen and Huang [3] proved that a quasi-transitive digraph is strong if and only if $Q=T[H_{1},\dots,H_{t}],$ where $T$ is a strong semicomplete digraph and each $H_{i}$ is a non-strong quasi-transitive digraph or has just one vertex. Thus, a special case of the above problem is to characterize strong quasi-transitive digraphs with a good decomposition. This would generalize Theorem 1.2 as well.

We believe that these characterizations will confirm Conjecture 1.1 for the classes of quasi-transitive digraphs and digraphs $T[H_{1},\dots,H_{t}],$ where $T$ is strong semicomplete. In the absence of the characterizations, it would still be interesting to confirm the conjecture at least for quasi-transitive digraphs.

In Lemma 3.2, we show that $G\square H$ contains a pair of arc-disjoint strong spanning subdigraphs when $G\cong H$. However, the following result implies Lemma 3.2 cannot be extended to the case that $G\not\cong H$, since it is not hard to show that the Cartesian product digraph of any two cycles has a pair of arc-disjoint strong spanning subdigraphs if and only if it has a pair of arc-disjoint Hamiltonian cycles.

Theorem 4.1

[13]** The Cartesian product $\overrightarrow{C_{p}}\square\overrightarrow{C_{q}}$ is Hamiltonian if and only if there are non-negative integers $d_{1},d_{2}$ for which $d_{1}+d_{2}=\gcd(p,q)\geq 2$ and $\gcd(p,d_{1})=\gcd(q,d_{2})=1$.

However, Lemma 3.2 could hold for the case that $G\not\cong H$ if we add other conditions. As shown in Lemma 3.3, we know $G\square H$ contains a pair of arc-disjoint strong spanning subdigraphs when one of $G$ and $H$ contains a pair of arc-disjoint strong spanning subdigraphs. So the following open question is interesting: for any two strong digraphs $G$ and $H$, neither of which contain a pair of arc-disjoint strong spanning subdigraphs, under what condition the product digraph $G\square H$ contains a pair of arc-disjoint strong spanning subdigraphs?

Furthermore, we may also consider the following more challenging question: under what conditions the product digraph $G\square H~{}(G\boxtimes H)$ has more (than two) arc-disjoint strong spanning subdigraphs?