The Hamilton–Waterloo Problem with C₁₆-Factors and C_m-Factors for Odd m

Abstract

The Hamilton–Waterloo problem is a problem of graph factorization. The Hamilton–Waterloo problem HWP$(H;m,n;\alpha ,\beta )$ asks for a two-factorization of a graph H containing $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors. Let $K_v^{*}$ denote the complete graph $K_v$ if v is odd and $K_v$ minus a one-factor if v is even. In this paper, we completely solve the Hamilton–Waterloo problem HWP$(K_v^{*};m,16;\alpha ,\beta )$ for odd $m\ge 9$ and $\alpha \ge 15$.

1. Introduction

A central theme in combinatorics and related areas is the decomposition of large discrete objects into simpler or smaller ones. Usually, these simpler or smaller objects are given in advance as needed and have some special properties such as symmetry and uniformity. In this paper, we will focus on a problem of graph factorization. We assume that the reader is familiar with basic concepts in graph theory and design theory, and refers to [1,2] for further details. In this paper, every graph will be simple. In general, the vertex-set and the edge-set of a graph H are denoted by $V(H)$ and $E(H)$, respectively. We denote the cycle of length k by $C_k$ and the complete graph on n vertices by $K_n$. We use $K_u\left[g\right]$ to denote the complete u-partite graph with u parts of size g. In fact, $K_u\left[1\right]$ is a complete graph $K_u$ and the graph $K_u\left[2\right]$ is $K_{2u}$ minus a one-factor. These graphs are all regular graphs and each of them possesses highly symmetric properties. A factor of H is a spanning subgraph of H whose vertex-set is exactly V(H). We call it a G-factor if its connected components are isomorphic to G. A G-factorization of H is a set of edge-disjoint G-factors of H whose edge-sets partition E(H). A $C_k$-factorization of H is a partition of E(H) into $C_k$-factors.

For the existence of a $C_k$-factorization of $K_u$, Ray-Chadhuri, Wilson [3], and Lu [4] independently proved the existence for the case of $k=3$. For the other cases, the necessary conditions of the existence of a $C_k$-factorization of $K_u$ are also sufficient, see [5,6,7]. The existence problem for a $C_k$-factorization of $K_u\left[2\right]$ has been solved, see [8,9]. Finally, Liu [10,11] completely solved the existence of a $C_k$-factorization of $K_u\left[g\right]$.

Theorem 1.

There exists a $C_k$-factorization of $K_u\left[g\right]$ if and only if $g(u-1)\equiv 0(\mathrm{mod}2)$, $gu\equiv 0(\mathrm{mod}k)$, k is even when $u=2$, and $(k,u,g)\notin \{(3,3,2),(3,6,2),(3,3,6),(6,2,6)\}$.

As usual, an r-regular factor is called an r-factor. In particular, a two-factor is a collection of vertex-disjoint cycles. A two-factorization of a graph H is a partition of $E(H)$ into two-factors. The Hamilton–Waterloo problem HWP$(H;m,n;\alpha ,\beta )$ asks for a two-factorization of a specified graph H containing $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors. Let $K_v^{*}$ denote the complete graph $K_v$ if v is odd and $K_v$ minus a one-factor if v is even. We denote a solution to HWP$(K_v^{*};m,n;\alpha ,\beta )$ by HW$(v;m,n;\alpha ,\beta )$. Also, we use HWP$(v;m,n)$ to denote the set of $(\alpha ,\beta )$ for which an HW$(v;m,n;\alpha ,\beta )$ exists. The necessary conditions for the existence of an HW$(v;m,n;\alpha ,\beta )$ are shown so that $m|v$ when $\alpha >0$, $n|v$ when $\beta >0$ and $\alpha +\beta =\lfloor \frac{v-1}{2}\rfloor$. Theorem 1 indicates that the existence of an HW$(v;m,n;\alpha ,\beta )$ has been completely solved when $\alpha \beta =0$.

For small values of m and n, the known results of the Hamilton–Waterloo problem are as follows. A complete solution for the existence of an HW$(v;3,n;\alpha ,\beta )$ in the cases $n\in \{4,5,7\}$ is given in [12,13,14,15,16,17]. For the cases $(m,n)\in \{(3,15),(5,15),(4,6),(4,8),(4,16)$, $(8,16)\}$, see [12]. Kamin [18] showed that the necessary conditions for the existence of an HW$(v;3,9;\alpha ,\beta )$ are also sufficient, apart from the exceptional case $\beta =1$. Asplund et al. [19] constructed many infinite classes of HW$(v;3,3x;\alpha ,\beta )$s.

The existence of an HW$(v;4,m;\alpha ,\beta )$ for odd $m\ge 3$ has been solved with some possible exceptions, see [16,17,20]. Fu and Huang [21] give a complete solution for an HW$(v;4,m;\alpha ,\beta )$ for even $m\ge 4$.

Theorem 2

([16,17,20,21]). $(\alpha ,\beta )\in$ HWP$(v;4,m)$ for $m\ge 3$ if and only if $\alpha ,\beta \ge 0$ and $\alpha +\beta =\lfloor \frac{v-1}{2}\rfloor$, except possibly when $m\ge 5$ is odd, $v=8m$, and $\alpha =2$.

Wang and Cao [22] considered the Hamilton–Waterloo problem with $C_8$-factors and $C_m$-factors and gave the following results.

Theorem 3

([22]). $(\alpha ,\beta )\in$ HWP$(8mt;8,m)$ for $m\ge 3$ and $t\ge 1$ if and only if $\alpha ,\beta \ge 0$ and $\alpha +\beta =4mt-1$, except possibly when $\alpha \in \{1,2\}$ and $mt$ is odd or $\alpha \in \{1,2,4,5,6\}$ and $mt$ is even.

Bryant et al. [23,24] completely solved the Hamilton–Waterloo problem for bipartite two-factors. Buratti and Danziger [25] as well as Merola and Traetta [26] focused on infinitely many cyclic solutions to the Hamilton–Waterloo problem with odd length cycles. Dinitz and Ling [27] as well as Lei and Shen [28] gave an analysis of the Hamilton–Waterloo problem for Hamilton cycles and triangle factors. Wang, Lu, and Cao [29] considered the existence of an HW$(k(2kt+1);k,2kt+1;\alpha ,\beta )$ for $t\ge 1$ and odd $k\ge 3$. For the case of two cycles sizes $m,n$ of different parity on the Hamilton–Waterloo problem, Keranen and Pastine [30] mainly focused on the case $m|n$ and $(m,n)=(2^{l}x,n)$ with odd $x,n$, and gcd$(x,n)\ge 3$. Burgess, Danziger, and Traetta studied the Hamilton–Waterloo problem in detail, see [31,32,33,34]. In 2022, Burgess et al. [35] made further progress when m and n are not coprime in two regards. In 2023, [36] presented a survey of constructive methods for the Hamilton–Waterloo problem which have allowed recent progress. The readers can have a comprehensive understanding of this problem.

In this paper, we consider the remaining situation of the Hamilton–Waterloo problem. We will focus on the existence of an HW$(16mt;16,m;\alpha ,\beta )$ for odd m and give the following main result.

Theorem 4.

For any odd $m\ge 9$, $(\alpha ,\beta )\in$ HWP$(16mt;16,m)$ if and only if $\alpha +\beta =8mt-1$, where $\alpha \ge 0$, $\beta \ge 0$, and $t\ge 1$, except possibly when $\alpha \in [1,6]$ and t is odd or $\alpha \in [1,6]\cup [8,14]$ and t is even.

2. Preliminary

In this section, we introduce some necessary definitions, notations, and known results which will be used later.

To begin with, we introduce the definition of a Cayley graph. Let $\mathsf{\Gamma }$ be a finite additive group and let S be a subset of $\mathsf{\Gamma }\setminus \{0\}$ closed under taking additive inverses. The Cayley graph over $\mathsf{\Gamma }$ with connection set S, denoted by $\mathrm{Cay}(\mathsf{\Gamma },S)$, is the graph with vertex-set $\mathsf{\Gamma }$ and edge-set $E(\mathrm{Cay}(\mathsf{\Gamma },S))=\{(a,b)|a,b\in \mathsf{\Gamma },a-b\in S\}$. For our constructions, we need the following results on a $C_m$-factorization or a $C_n$-factorization of $\mathrm{Cay}(Z_m\times Z_n,S)$.

Lemma 1

([17,22]). $(1)$ Let $m,n\ge 3$, let $a\in Z_n$ satisfying $|\pm \{0,a,2a\}|=5$, and let $\gcd (i,m)=1$. There exist five $C_m$-factors which form a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_n,\{\pm i\}\times (\pm \{0,a,2a\}))$.

$(2)$ Let $m\ge 3$ be odd, let $n\ge 4$ be even, and let $a,b\in Z_n$ with $|\pm \{a,b,a+b\}|=6$. There exist six $C_m$-factors which form a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_n,\{\pm i\}\times (\pm \{a,b,a+b\}))$ with $\gcd (i,m)=1$.

$(3)$ Let $m\ge 3$ be odd, let $n\ge 4$ be even, and let $1\le d<n$. There exist three $C_m$-factors which form a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_n,\{\pm i\}\times \{0,\pm d\})$ with $\gcd (i,m)=1$.

$(4)$ Let $n\ge 4$ be even and let $d=0$ $(m\ge 3)$ or $d=n/2$ $(m\ge 4$ is even). There exists a $C_m$-factor which forms a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_n,\{\pm i\}\times \{d\})$ with $\gcd (i,m)=1$.

$(5)$ Let $m\ge 3$, let $n\ge 4$ be even and let $0<d<n$ be coprime to n. There exist two $C_n$-factors which form a $C_n$-factorization of $\mathrm{Cay}(Z_m\times Z_n,\{\pm i\}\times \{\pm d\})$ with $\gcd (i,m)=1$.

Next, we introduce the concept of the wreath product of two graphs. If both G and H are graphs, the wreath product $G\wr H$ of G and H has a vertex-set $V(G)\times V(H)$ in which $(u_1,v_1)(u_2,v_2)\in E(G\wr H)$ whenever $u_1{u}_2\in E(G)$ or, $u_1=u_2$ and $v_1{v}_2\in E(H)$. For brevity, we denote $C_m\wr \overline{K_n}$ by $C_m\left[n\right]$, where $\overline{K_n}$ is the complement of $K_n$. We will give some known results and constructions that will be used later.

Theorem 5

([6,37]). For $m\ge 3$ and $n\ge 1$, a $C_m$-factorization of $C_m\left[n\right]$ exists, except for $(m,n)=(3,6)$ and $(m,n)\in \{(l,2)|l\ge 3$ is odd}.

Theorem 6

([38]). For $m\ge 3$ and $n\ge 1$, there exists a $C_{mn}$-factorization of $C_m\left[n\right]$.

Theorem 7

([16]). The graph $C_m\left[4\right]$ can be decomposed into α $C_4$-factors and $4-\alpha$ $C_m$-factors for $m\ge 3$ and $\alpha \in \{0,2,4\}$.

Construction 1

([22]). If $(\alpha ,\beta )\in$ HWP$(C_m\left[n\right];m,n)$, then $(\alpha ,\beta +\lfloor \frac{n-1}{2}\rfloor )\in$ HWP$(C_m\wr K_n;m,n)$.

Construction 2

([17]). If there exists an HW$(K_u\left[g\right];m,n;\alpha ,\beta )$ and an HW$(g;m,n;{\alpha }^{\prime },{\beta }^{\prime })$, then an HW$(gu;m,n;\alpha +{\alpha }^{\prime },\beta +{\beta }^{\prime })$ exists.

Construction 3.

If $C_m\left[n\right]$ can be decomposed into $\alpha$ $C_n$-factors and $n-\alpha$ $C_m$-factors, and there exists a $C_{nw}$-factorization of $C_n\left[w\right]$ and a $C_m$-factorization of $C_m\left[w\right]$, then $C_m\left[nw\right]$ can be decomposed into $w\alpha$ $C_{nw}$-factors and $w(n-\alpha )$ $C_m$-factors.

Proof. 

The graph $C_m\left[n\right]$ can be decomposed into $\alpha$ $C_n$-factors and $n-\alpha$ $C_m$-factors. Then, we give each vertex weight w to obtain $\alpha$ $C_n\left[w\right]$-factors and $n-\alpha$ $C_m\left[w\right]$-factors. Each $C_n\left[w\right]$ $(C_m\left[w\right])$ can be partitioned into w $C_{nw}$-factors ($C_m$-factors). Finally, each $C_m\left[nw\right]$-factor can be decomposed into $w\alpha$ $C_{nw}$-factors and $w(n-\alpha )$ $C_m$-factors.  □

3. Decompositions of Some Cayley Graphs

In this section, we will give some new decompositions of Cayley graphs. For brevity, we will denote the vertex $(x,y)$ by $x_y$.

Lemma 2.

Let $m\ge 3$ be odd and $n\equiv 0(\mathrm{mod}4)$. The graph $\mathrm{Cay}(Z_m\times Z_n,\{\pm 1\}\times \{\pm \frac{n}{4},\frac{n}{2}\})$ can be decomposed into three $C_m$-factors.

Proof. 

Let

	$C_1^1=(0_0,1_{\frac{n}{4}},2_{\frac{n}{2}},3_{\frac{n}{4}},4_{\frac{n}{2}},\dots ,{(m-2)}_{\frac{n}{4}},{(m-1)}_{\frac{n}{2}})$,
	$C_1^2=(0_{\frac{n}{4}},1_0,2_{-\frac{n}{4}},3_0,4_{-\frac{n}{4}},\dots ,{(m-2)}_0,{(m-1)}_{-\frac{n}{4}})$,
	$C_2^1=(0_0,1_{-\frac{n}{4}},2_{\frac{n}{4}},3_{-\frac{n}{4}},4_{\frac{n}{4}},\dots ,{(m-2)}_{-\frac{n}{4}},{(m-1)}_{\frac{n}{4}})$,
	$C_2^2=(0_{-\frac{n}{4}},1_0,2_{\frac{n}{2}},3_0,4_{\frac{n}{2}},\dots ,{(m-2)}_0,{(m-1)}_{\frac{n}{2}})$,
	$C_3^1=(0_0,1_{\frac{n}{2}},2_{-\frac{n}{4}},3_{\frac{n}{2}},4_{-\frac{n}{4}},\dots ,{(m-2)}_{\frac{n}{2}},{(m-1)}_{-\frac{n}{4}})$,
	$C_3^2=(0_{-\frac{n}{4}},1_{\frac{n}{4}},2_0,3_{\frac{n}{4}},4_0,\dots ,{(m-2)}_{\frac{n}{4}},{(m-1)}_0)$.

Let ${\mathcal{B}}_i=\{C_i^j+(0,s),C_i^j+(0,s+\frac{n}{2})|1\le j\le 2,0\le s\le \frac{n}{4}-1\}$, $1\le i\le 3$. Each ${\mathcal{B}}_i$ has $2\times 2\times \frac{n}{4}=n$ cycles with length m and $V({\mathcal{B}}_i)=Z_m\times Z_n$, thus it is a $C_m$-factor. By counting the edges of ${\cup }_{i=1}^3{\mathcal{B}}_i$, we obtain that ${\cup }_{i=1}^{3}E({\mathcal{B}}_i)$ coincides with the edge-set of the Cayley graph $\mathrm{Cay}(Z_m\times Z_n,\{\pm 1\}\times \{\pm \frac{n}{4},\frac{n}{2}\})$. So, this Cayley graph can be decomposed into three $C_m$-factors.  □

Lemma 3.

Let $l\ge 3$ and $m\ge 2^{l-1}+1$ be odd. The graph $\mathrm{Cay}(Z_m\times Z_{2^l},\{\pm 1\}\times \{\pm 1,2^{l-1}\})$ can be partitioned into two $C_{2^l}$-factors and a $C_m$-factor.

Proof. 

Let

$C_1=(0_0,{(m-1)}_1,0_2,{(m-1)}_3,\dots ,0_{2^l-2},{(m-1)}_{2^l-1})$,

$C_2=(0_1,1_{1+2^{l-1}},2_1,3_{1+2^{l-1}},\dots ,{(2^{l-1}-2)}_1,{(2^{l-1}-1)}_{1+2^{l-1}},{(2^{l-1})}_{2^{l-1}},{(2^{l-1}-1)}_0,{(2^{l-1}-2)}_{2^{l-1}},{(2^{l-1}-3)}_0,\dots ,2_{2^{l-1}},1_0)$,

$C_3=({(m-1)}_0,0_1,{(m-1)}_2,0_3,\dots ,{(m-1)}_{2^l-2},0_{2^l-1})$,

$C_4=(0_0,1_{2^{l-1}},2_{2^{l-1}-1},3_{2^{l-1}-2},\dots ,{(2^{l-1}-2)}_3,{(2^{l-1}-1)}_2,{(2^{l-1})}_1,{(2^{l-1}-1)}_{2^{l-1}+1},{(2^{l-1}-2)}_{2^{l-1}+2},\dots ,2_{2^l-2},1_{2^l-1})$.

(1) For $m=2^{l-1}+1$, let ${\mathcal{B}}_1=\{(0_0,1_1,2_2,\dots ,{(2^{l-1})}_{2^{l-1}})+(0,h)|h\in Z_{2^l}\}$. It contains $2^l$ cycles with a length of m and $V({\mathcal{B}}_1)=Z_m\times Z_{2^l}$, then ${\mathcal{B}}_1$ is a $C_m$-factor.

Let ${\mathcal{B}}_2=\{C_1,C_2+(0,2i)|0\le i\le 2^{l-1}-1\}$ and ${\mathcal{B}}_3=\{C_3,C_4+(0,2i)|0\le i\le 2^{l-1}-1\}$. Each of them has m cycles with a length of $2^l$ and its vertex-set is $Z_m\times Z_{2^l}$, then ${\mathcal{B}}_2$ and ${\mathcal{B}}_3$ are two $C_{2^l}$-factors. By counting ${\cup }_{i=1}^{3}E({\mathcal{B}}_i)$, we obtain that it coincides with the edge-set of the Cayley graph.

(2) For $m\ge 2^{l-1}+3$, let

${\mathcal{B}}_1=\{(0_0,1_1,\dots ,{(2^{l-1})}_{2^{l-1}},{(2^{l-1}+1)}_0,{(2^{l-1}+2)}_{2^{l-1}},{(2^{l-1}+3)}_0,{(2^{l-1}+4)}_{2^{l-1}},\cdots ,$ ${(m-2)}_0,{(m-1)}_{2^{l-1}})+(0,h)|h\in Z_{2^l}\}$,

${\mathcal{B}}_2=\{C_1,C_2+(0,2i),({(j+1)}_0,j_1,{(j+1)}_2,j_3,\dots ,{(j+1)}_{2^l-2},j_{2^l-1})|0\le i\le 2^{l-1}-1,2^{l-1}\le j\le m-2\}$,

${\mathcal{B}}_8=\{C_3,C_4+(0,2i),(j_0,{(j+1)}_1,j_2,{(j+1)}_3,\dots ,j_{2^l-2},{(j+1)}_{2^l-1})|0\le i\le 2^{l-1}-1,2^{l-1}\le j\le m-2\}$.

Similarly to the above case, we obtain that ${\mathcal{B}}_1$ is a $C_m$-factor and ${\mathcal{B}}_2,{\mathcal{B}}_3$ are two $C_{2^l}$-factors. We check that ${\cup }_{i=1}^{3}E({\mathcal{B}}_i)=E(\mathrm{Cay}(Z_m\times Z_{2^l},\{\pm 1\}\times \{\pm 1,2^{l-1}\}))$.  □

Lemma 4.

Let $m\ge 3$ and $l\ge 3$. The graph $\mathrm{Cay}(Z_m\times Z_{2^l},\{\pm 1\}\times \{2^{l-1}\})\cup m{K}_{2^l}$ can be partitioned into $2^{l-1}$ $C_{2^l}$-factors and a one-factor.

Proof. 

Note that the graph $m{K}_{2^l}$ is equivalent to the Caylay graph $\mathrm{Cay}(Z_m\times Z_{2^l},\{0\}\times (Z_{2^l}\setminus \{0\}))$. By Theorem 1, there exists a $C_{2^l}$-factorization of the graph $K_{2^{l-1}}\left[2\right]$. Let $(e_1,e_2)=(0,1),(e_3,e_4)=(1+2^{l-1},2+2^{l-1}),(e_{2t+1},e_{2t+2})=(t,t+1+2^{l-1}),2\le t\le 2^{l-1}-2,$ $(e_{2^l-1},e_{2^l})=(2^{l-1}-1,2^{l-1})$. Without loss of generality, let $\{\{e_1,e_2\},\{e_3,e_4\},\dots ,\{e_{2^l-1},e_{2^l}\}\}$ be the group set of $K_{2^{l-1}}\left[2\right]$. There are $2^{l-1}-1$ $C_{2^l}$-factors of $K_{2^{l-1}}\left[2\right]$, denoted by $(b_{s1},b_{s2},\dots ,$ $b_{s,2^l})$ for $1\le s\le 2^{l-1}-1$.

We first construct the required $2^{l-1}$ $C_{2^l}$-factors, each of which has exactly m cycles with a length of $C_{2^l}$. Let $C_s=(0_{b_{s1}},0_{b_{s2}},\dots ,0_{b_{s,2^l}})$, $1\le s\le 2^{l-1}-1$, and

$$C_{2^{l-1}}=(0_{e_1},0_{e_2},1_{e_3},1_{e_4},0_{e_5},0_{e_6},1_{e_7},1_{e_8},\dots ,0_{e_{2^l-3}},0_{e_{2^l-2}},1_{e_{2^l-1}},1_{e_{2^l}}).$$

For $1\le i\le 2^{l-1}$, the set of the subscripts of $C_i$ is actually $Z_{2^l}$, thus, each $C_i$ can generate a $C_{2^l}$-factor by $(+1(\mathrm{mod}m),-)$. In other words, $\{C_i+(l,0)|l\in Z_m\}$ is a $C_{2^l}$-factor.

In the original graph, in addition to the edges in the cycles above, there are still some edges left, that is, $\{(1_{e_{4p-2}},0_{e_{4p-1}})+(l,0),(0_{e_{4p}},1_{e_{4p+1}})+(l,0)|l\in Z_m,1\le p\le 2^{l-2},e_{2^l+1}=e_1\}.$ The set of vertices on these edges is $Z_m\times Z_{2^l}$, so this set of edges forms a one-factor.  □

Now, we construct two special one-factorizations of $K_{16}$ with the vertex-set $Z_{16}$ whose 15 one-factors are listed below for the following lemmas.

$I_1=\{(0,1),(3,6),(4,5),(7,10),(8,9),(11,14),(12,13),(15,2)\}$,

$I_3=\{(0,2),(6,1),(13,3),(7,9),(5,11),(15,12),(8,10),(14,4)\}$,

$I_5=\{(0,4),(10,1),(11,3),(9,2),(12,8),(14,5),(15,7),(13,6)\}$,

$I_7=\{(0,5),(3,7),(9,4),(2,10),(12,1),(15,11),(13,8),(6,14)\}$,

$I_9=\{(0,6),(2,4),(8,3),(15,5),(9,11),(7,13),(1,14),(10,12)\}$,

$I_{11}=\{(0,15),(9,5),(11,12),(6,2),(8,7),(1,13),(3,4),(14,10)\}$,

$I_{13}=\{(0,9),(6,12),(5,2),(11,8),(14,7),(4,13),(15,1),(3,10)\}$,

$I_{14}=\{(9,6),(12,5),(2,11),(8,14),(7,4),(13,15),(1,3),(10,0)\}$,

$I_{15}=\{(0,11),(13,2),(12,7),(14,9),(3,5),(10,4),(15,6),(1,8)\}$,

$I_{2i}=\{(x+2i,y+2i):(x,y)\in I_{2i-1}\}$, $i=1,2,3$,

$I_{6+2i}=\{(x-2i,y-2i):(x,y)\in I_{5+2i}\}$, $i=1,2,3$.

Note that $I_{13}\cup I_{14}$ can form a 16-cycle.

$I_1^{\prime }=\{(0,2),(1,3),(5,4),(6,8),(7,9),(11,10),(15,13),(12,14)\}$,

$I_2^{\prime }=\{(2,1),(3,5),(4,6),(8,7),(9,11),(10,15),(13,12),(14,0)\}$,

$I_3^{\prime }=\{(0,4),(2,3),(6,1),(5,7),(10,8),(14,13),(9,15),(11,12)\}$,

$I_4^{\prime }=\{(4,2),(3,6),(1,5),(7,10),(8,14),(13,9),(15,11),(12,0)\}$,

$I_5^{\prime }=\{(0,5),(8,1),(7,3),(9,2),(15,4),(14,10),(12,6),(11,13)\}$,

$I_6^{\prime }=\{(5,8),(1,7),(3,9),(2,15),(4,14),(10,12),(6,11),(13,0)\}$,

$I_7^{\prime }=\{(0,6),(2,8),(3,12),(9,14),(5,10),(1,15),(7,13),(4,11)\}$,

$I_8^{\prime }=\{(6,2),(8,3),(12,9),(14,5),(10,1),(15,7),(13,4),(11,0)\}$,

$I_9^{\prime }=\{(0,7),(12,2),(13,3),(14,6),(10,4),(9,1),(11,5),(15,8)\}$,

$I_{10}^{\prime }=\{(7,12),(2,13),(3,14),(6,10),(4,9),(1,11),(5,15),(8,0)\}$,

$I_{11}^{\prime }=\{(0,10),(2,11),(3,15),(7,14),(6,13),(5,9),(1,12),(4,8)\}$,

$I_{12}^{\prime }=\{(0,1),(7,2),(12,4),(10,3),(13,5),(11,14),(8,9),(15,6)\}$,

$I_{13}^{\prime }=\{(6,7),(13,8),(2,10),(0,9),(3,11),(1,4),(14,15),(5,12)\}$,

$I_{14}^{\prime }=\{(0,3),(13,1),(7,4),(14,2),(8,11),(5,6),(12,15),(9,10)\}$,

$I_{15}^{\prime }=\{(10,13),(7,11),(1,14),(8,12),(2,5),(15,0),(6,9),(3,4)\}$.

Note that $I_{2i-1}^{\prime }\cup I_{2i}^{\prime }$ can form a 16-cycle for $1\le i\le 5$.

For an integer $m\ge 2$, $mH$ denotes m vertex-disjoint copies of a graph H. For brevity, we use $m{I}_k$ (or $m{I}_k^{\prime })$ to denote the graph with the vertex-set $Z_m\times Z_{16}$ and the edge-set $\{(j_a,j_b)|j\in Z_m,(a,b)\in I_k($or $I_k^{\prime }),a\ne b\}$ for $1\le k\le 15$. Similarly, $m{K}_n$ denotes the graph with the vertex-set $Z_m\times Z_n$ and the edge-set $\{(j_a,j_b)|j\in Z_m,(a,b)\in E(K_n)\}$.

Lemma 5.

Let $m\ge 3$ and $i\in \{2,4,6\}$. There exist two $C_{16}$-factors which form a $C_{16}$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{i\})\cup m{I}_{i-1}\cup m{I}_i$.

Proof. 

Let $C_1^2=(0_0,0_1,1_3,1_6,0_4,0_5,1_7,1_{10},0_8,0_9,1_{11},1_{14},0_{12},0_{13},1_{15},1_2),$

$$C_1^4=(0_0,0_2,1_6,1_1,0_{13},0_3,1_7,1_9,0_5,0_{11},1_{15},1_{12},0_8,0_{10},1_{14},1_4),$$

$$C_1^6=(0_0,0_4,1_{10},1_1,0_{11},0_3,1_9,1_2,0_{12},0_8,1_{14},1_5,0_{15},0_7,1_{13},1_6).$$

For each $i\in \{2,4,6\}$, let $C_2^i=C_1^i+(0,i)$. Because the set of the subscripts of $C_t^i$ is $Z_{16}$, each ${\mathcal{B}}_t^i=\{C_t^i+(l,0)|l\in Z_m\}$ is a $C_{16}$-factor for $t=1,2$. Since $E({\mathcal{B}}_1^i)\cup E({\mathcal{B}}_2^i)=E(\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{i\})\cup m{I}_{i-1}\cup m{I}_i)$, these two $C_{16}$-factors can form a $C_{16}$-factorization of the graph.  □

Lemma 6.

Let $m\ge 3$ and $i\in \{2,4,6\}$. There exist two $C_{16}$-factors which is a $C_{16}$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{-i\})\cup m{I}_{6+i-1}\cup m{I}_{6+i}$.

Proof. 

Let $C_1^2=(0_0,0_5,1_3,1_7,0_9,0_4,1_2,1_{10},0_{12},0_1,1_{15},1_{11},0_{13},0_8,1_6,1_{14})$,

$$C_1^4=(0_0,0_6,1_2,1_4,0_8,0_3,1_{15},1_5,0_9,0_{11},1_7,1_{13},0_1,0_{14},1_{10},1_{12}),$$

$$C_1^6=(0_0,0_{15},1_9,1_5,0_{11},0_{12},1_6,1_2,0_8,0_7,1_1,1_{13},0_3,0_4,1_{14},1_{10}).$$

It is similar to the above lemma, let $C_2^i=C_1^i+(0,-i)$ for any $i\in \{2,4,6\}$. We have that each $\{C_t^i+(l,0)|l\in Z_m\}$ is a $C_{16}$-factor for $t=1,2$, and they form a $C_{16}$-factorization.  □

Lemma 7.

Let $m\ge 3$. The graph $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{8\})\cup m({\cup }_{i=13}^{15}{I}_i)$ can be decomposed into two $C_{16}$-factors and a one-factor.

Proof. 

Let $C_1=(0_0,0_9,0_6,0_{12},0_5,0_2,0_{11},0_8,0_{14},0_7,0_4,0_{13},0_{15},0_1,0_3,0_{10}),$

$$C_2=(0_0,0_{11},1_3,1_5,0_{13},0_2,1_{10},1_4,0_{12},0_7,1_{15},1_6,0_{14},0_9,1_1,1_8).$$

Similarly, each ${\mathcal{B}}_t=\{C_t+(l,0)|l\in Z_m\}$ is a $C_{16}$-factor for $t=1,2$ because the set of the subscripts of $C_t$ is $Z_{16}$. Let $I=\{({(j+1)}_{11},j_3),(j_5,{(j+1)}_{13}),({(j+1)}_2,j_{10}),(j_4,{(j+1)}_{12}),({(j+1)}_7,j_{15}),$ $(j_6,{(j+1)}_{14}),({(j+1)}_9,j_1),(j_8,{(j+1)}_0)|j\in Z_m\}$. It is a one-factor since $V(I)=Z_m\times Z_{16}$. We check that $E(\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{8\})\cup m({\cup }_{i=13}^{15}{I}_i))=E({\mathcal{B}}_1)\cup E({\mathcal{B}}_2)\cup E(I)$. Then, we obtain the conclusion.  □

Lemma 8.

Let $m\ge 3$. The graph $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{8\})\cup m{I}_{11}^{\prime }$ can be partitioned into a $C_{16}$-factor and a one-factor.

Proof. 

Let $C=(0_0,0_{10},1_2,1_{11},0_3,0_{15},1_7,1_{14},0_6,0_{13},1_5,1_9,0_1,0_{12},1_4,1_8).$ Since the set of the subscripts of C is $Z_{16}$, we obtain that $\mathcal{B}=\{C+(l,0)|l\in Z_m\}$ is a $C_{16}$-factor of this graph. Let $I=\{({(j+1)}_{10},j_2),(j_{11},{(j+1)}_3),({(j+1)}_{15},j_7),(j_{14},{(j+1)}_6),({(j+1)}_{13},j_5),(j_9,{(j+1)}_1),({(j+1)}_{12},j_4),$ $(j_8,{(j+1)}_0)|j\in Z_m\}$. It is a set of edges and $V(I)=Z_m\times Z_{16}$, so it is a one-factor. We check that $E(\mathcal{B})\cup E(I)$ coincides with the edge-set of the graph $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{8\})\cup m{I}_{11}^{\prime }$ by counting the number of edges.  □

Lemma 9.

Let $m\ge 3$. The graph $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 6\})\cup m({\cup }_{i=12}^{15}{I}_i^{\prime })$ can be decomposed into four $C_{16}$-factors.

Proof. 

Let $C_1=(0_0,0_1,1_7,1_2,0_{12},0_4,1_{10},1_3,0_{13},0_5,1_{11},1_{14},0_8,0_9,1_{15},1_6),$

$$C_2=(0_6,0_7,1_{13},1_8,0_2,0_{10},1_0,1_9,0_3,0_{11},1_1,1_4,0_{14},0_{15},1_5,1_{12}),$$

$$C_3=(0_0,0_3,1_{13},1_1,0_7,0_4,1_{14},1_2,0_8,0_{11},1_5,1_6,0_{12},0_{15},1_9,1_{10}),$$

$$C_4=(0_{10},0_{13},1_7,1_{11},0_1,0_{14},1_8,1_{12},0_2,0_5,1_{15},1_0,0_6,0_9,1_3,1_4).$$

Let ${\mathcal{B}}_i=\{C_t+(l,0)|l\in Z_m\}$ for $1\le t\le 4$. Since the subscripts of $C_t$ form the set $Z_{16}$, each ${\mathcal{B}}_i$ is a $C_{16}$-factor. By counting the edges of ${\cup }_{i=1}^4{\mathcal{B}}_i$, we obtain the required design.  □

4. Main Results

In this section, we will prove our main results.

Lemma 10.

For odd $m\ge 9$ and $r\in \{0,2,4,6,8,16\}$, $(r,16-r)\in$ HWP$(C_m\left[16\right];16,m)$.

Proof. 

We consider the four following cases.

Case 1: $r=0,8,16$.

By Theorem 7, the graph $C_m\left[4\right]$ can be decomposed into $\frac{r}{4}$ $C_4$-factors and $4-\frac{r}{4}$ $C_m$-factors for $m\ge 3$. By Theorems 5 and 6, two graphs $C_m\left[4\right]$ and $C_4\left[4\right]$ can be partitioned into four $C_m$-factors and four $C_{16}$-factors, respectively. Then, we obtain the conclusion by applying Construction 3.

Case 2: $r=2$.

We obtain two $C_{16}$-factors from a $C_{16}$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 5\})$ by Lemma 1(5). The required fourteen $C_m$-factors can be obtained through three parts. The graph $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 1,\pm 2\})$ can be decomposed into five $C_m$-factors by Lemma 1(1). Similarly, we consider two Cayley graphs $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{3,6,7\}))$ and $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 4,8\})$. They can be partitioned into six $C_m$-factors and three $C_m$-factors by Lemma 1(2) and Lemma 2, respectively.

Case 3: $r=4$.

Four $C_{16}$-factors are given from a $C_{16}$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{5,7\}))$ by Lemma 1(5). Six of the twelve required $C_m$-factors can be obtained from the decomposition of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{1,2,3\}))$ by Lemma 1(2). The graph $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 6\})$ can be decomposed into three $C_m$-factors by Lemma 1(3). The last three $C_m$-factors come from the decomposition of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 4,8\})$ by Lemma 2.

Case 4: $r=6$.

A $C_{16}$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{5,7\}))$ generates four $C_{16}$-factors by Lemma 1(5). The graph $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 1,8\})$ can be decomposed into two $C_{16}$-factors and a $C_m$-factor by Lemma 3. Now, we have the six required $C_{16}$-factors. The nine remaining $C_m$-factors are listed below. Six of them can be obtained from a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{2,4,6\}))$ by Lemma 1(2). The last three come from the decomposition of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 3\})$ by Lemma 1(3).  □

Lemma 11.

For odd $m\ge 9$ and $7\le r\le 23$, the graph $C_m\wr K_{16}$ can be partitioned into r$C_{16}$-factors, $23-r$ $C_m$-factors and a one-factor.

Proof. 

Let the vertex-set be $Z_m\times Z_{16}$. We distinguish 12 cases as shown below.

Case 1: $r\in \{7,9,11,13,15,23\}$.

By Theorem 1 and Lemma 10, we have $(0,7)\in$ HWP$(16;m,16)$ and $(16-r_1,r_1)\in$ HWP$(C_m\left[16\right];m,16)$ for odd $m\ge 9$ and $r_1\in \{0,2,4,6,8,16\}$, respectively. Applying Construction 1, we obtain $(16-r_1,r_1+7)\in$ HWP$(C_m\wr K_{16};m,16)$.

Case 2: $r=8$.

$\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{8\})\cup m{K}_{16}$ can be decomposed into eight $C_{16}$-factors and a one-factor by Lemma 4. $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{1,5,6\}))$ and $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{3,4,7\}))$ can be partitioned into twelve $C_m$-factors from Lemma 1(2). A $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 2\})$ can generate three $C_m$-factors by Lemma 1(3).

Case 3: $r=10$.

$\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 6,8\})\cup m({\cup }_{i=11}^{15}{I}_i^{\prime })$ can be decomposed into five $C_{16}$-factors and a one-factor from Lemmas 8 and 9. Since $I_{2j-1}^{\prime }\cup I_{2j}^{\prime }$ can form a 16-cycle, we can obtain a $C_{16}$-factor for $1\le j\le 5$ from the graph $m(I_{2j-1}^{\prime }\cup I_{2j}^{\prime })$. In other words, we obtained ten $C_{16}$-factors and a one-factor from $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 6,8\})\cup m{K}_{16}$.

Two Cayley graphs $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{1,3,4\}))$ and $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{2,5,7\}))$ can be partitioned into 12 $C_m$-factors by Lemma 1(2). The last $C_m$-factor comes from a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0\})$ by Lemma 1(4).

Case 4: $r=12$.

$\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 6,8\})\cup m{K}_{16}$ can be decomposed into ten $C_{16}$-factors and a one-factor from the above case. By Lemma 1(5), we can obtain the two remaining $C_{16}$-factors from the decomposition of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 5\})$. Five $C_m$-factors come from a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 1,\pm 2\})$ by Lemma 1(1). $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{3,4,7\}))$ is precisely divided into six $C_m$-factors by Lemma 1(2).

Case 5: $r=14$.

The Cayley graph $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{8\})\cup m{K}_{16}$ can be divided into eight $C_{16}$-factors and a one-factor by Lemma 4. The last six $C_{16}$-factors come from a $C_{16}$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{3,5,7\}))$ by Lemma 1(5). In addition, $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{2,4,6\}))$ can be partitioned into six $C_m$-factors by Lemma 1(2). A $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 1\})$ contains exactly three $C_m$-factors by Lemma 1(3).

Case 6: $r=16$.

By Lemma 4, $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{8\})\cup m{K}_{16}$ can be decomposed into eight $C_{16}$-factors and a one-factor. The last eight $C_{16}$-factors originate from a $C_{16}$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{1,3,5,7\}))$ by Lemma 1(5). $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{2,4,6\}))$ can be divided into six $C_m$-factors from Lemma 1(2). The last $C_m$-factor comes from a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0\})$ by Lemma 1(4).

Case 7: $r=18$.

$\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 6,8\})\cup m{K}_{16}$ can be divided into ten $C_{16}$-factors and a one-factor from Case 3. Similarly, we obtain eight $C_{16}$-factors from $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{1,3,5,7\}))$ by Lemma 1(5). The five required $C_m$-factors originate from a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 2,\pm 4\})$ by Lemma 1(1).

Case 8: $r=20$.

$\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{2,4,6\}))\cup m({\cup }_{i=1}^{12}{I}_i)$ can be partitioned into twelve $C_{16}$-factors by Lemmas 5 and 6. $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{8\})\cup m({\cup }_{i=13}^{15}{I}_i)$ can be divided into two $C_{16}$-factors and a one-factor by Lemma 7. That is to say, $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 2,\pm 4,\pm 6,8\})\cup m{K}_{16}$ can be decomposed into fourteen $C_{16}$-factors and a one-factor. We can obtain the other six $C_{16}$-factors from $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{3,5,7\}))$ by Lemma 1(5) and obtain three $C_m$-factors from a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 1\})$ by Lemma 1(3).

Case 9: $r=22$.

$\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 2,\pm 4,\pm 6,8\})\cup m{K}_{16}$ can be decomposed into fourteen $C_{16}$-factors and a one-factor from the above case. The other eight $C_{16}$-factors can be obtained from $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{1,3,5,7\}))$ by Lemma 1(5). A $C_m$-factor originates from the decomposition of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0\})$ by Lemma 1(4).

Case 10: $r=17$.

Since $I_{2j-1}^{\prime }\cup I_{2j}^{\prime }$ form a 16-cycle, the graph $m(I_{2j-1}^{\prime }\cup I_{2j}^{\prime })$ is actually a $C_{16}$-factor for any $1\le j\le 5$. Two graphs $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 6\})\cup m({\cup }_{i=12}^{15}{I}_i^{\prime })$ and $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{3,5,7\}))$ can be decomposed into four $C_{16}$-factors and six $C_{16}$-factors by Lemma 9 and Lemma 1(5), respectively. We can obtain two $C_{16}$-factors and a $C_m$-factor from $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 1,8\})$ by Lemma 3 and five $C_m$-factors from a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 2,\pm 4\})$ by Lemma 1(1). The one-factor is $m{I}_{11}^{\prime }$.

Case 11: $r=19$.

By Lemmas 5 and 6, the graph $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{2,4,6\}))\cup m({\cup }_{i=1}^{12}{I}_i)$ can be divided into 12 $C_{16}$-factors. $m(I_{13}\cup I_{14})$ is a $C_{16}$-factor since $I_{13}\cup I_{14}$ can form a 16-cycle. The one-factor is $m{I}_{15}$. $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{\pm 1,8\})$ can be divided into two $C_{16}$-factors and a $C_m$-factor by Lemma 3. That is to say, we can obtain 15 $C_{16}$-factors, a $C_m$-factor, and a one-factor from $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{1,2,4,6\}\cup \{8\}))\cup m{K}_{16}$.

The remaining four $C_{16}$-factors and three $C_m$-factors come from the factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{5,7\}))$ and $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0,\pm 3\})$ by Lemma 1(5) and Lemma 1(3), respectively.

Case 12: $r=21$.

Similarly to the above case, we obtain 15 $C_{16}$-factors, a $C_m$-factor, and a one-factor from $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{1,2,4,6\}\cup \{8\}))\cup m{K}_{16}$. $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times (\pm \{3,5,7\}))$ can be decomposed into six $C_{16}$-factors by Lemma 1(5). The last $C_m$-factor originates from a $C_m$-factorization of $\mathrm{Cay}(Z_m\times Z_{16},\{\pm 1\}\times \{0\})$ by Lemma 1(4).  □

Proof of Theorem 4.

Let the vertex-set of $K_{16mt}$ be $Z_{mt}\times Z_{16}$. We distinguish the two following cases.

Case 1: t is odd.

The complete graph $K_{mt}$ on the vertex-set $Z_{mt}$ can be decomposed into $\frac{mt-1}{2}$ $C_m$-factors by Theorem 1. We give each vertex of $Z_{mt}$ a weight of 16 to obtain $mt{K}_{16}$ and $\frac{mt-1}{2}$ $C_m\left[16\right]$-factors, which are denoted by $P_i$, $1\le i\le \frac{mt-1}{2}$. Each $P_i$ has t $C_m\left[16\right]$s, denoted by $Q_{ij}$, $1\le j\le t$.

Let $0\le x\le \frac{mt-3}{2}$, we replace each $P_i$ $(1\le i\le x)$ with an HW$(C_m\left[16\right];16,m;16,0)$ and $P_i$ $(x+1\le i\le \frac{mt-3}{2})$ with an HW$(C_m\left[16\right];16,m;0,16)$ by Lemma 10.

For $P_{\frac{mt-1}{2}}$ and $1\le j\le t$, the graph $Q_{\frac{mt-1}{2},j}\cup m{K}_{16}$ can be partitioned into r $C_{16}$-factors, $23-r$ $C_m$-factors, and a one-factor for $7\le r\le 23$ by Lemma 11. We put them together to obtain r $C_{16}$-factors, $23-r$ $C_m$-factors, and a one-factor on the vertex-set $Z_{mt}\times Z_{16}$.

We finally obtain $\alpha =16x+r$ $C_{16}$-factors, $\beta =16\times (\frac{mt-3}{2}-x)+23-r=8mt-1-(16x+r)$ $C_m$-factors and a one-factor for $0\le x\le \frac{mt-3}{2}$ and $7\le r\le 23$. Here, the range for $\alpha$ is 7 to $8mt-1$.

Case 2: t is even.

$(1)$ $\alpha =7$.

We can obtain the conclusion by using Construction 2 with an HW$(16;$ $16,m;7,0)$ and an HW$(K_{mt}\left[16\right];16,m;0,8mt-8)$ from Theorem 1.

$(2)$ $\alpha \ge 15$.

The graph $K_{\frac{mt}{2}}\left[2\right]$ can be partitioned into $\frac{mt-2}{2}$ $C_m$-factors by Theorem 1. In other words, the graph $K_{mt}$ can be decomposed into $\frac{mt-2}{2}$ $C_m$-factors and a one-factor. Giving each vertex of the graph $K_{mt}$ weight 16, we obtain $\frac{mt-2}{2}$ $C_m\left[16\right]$-factors which are denoted by $P_i$, $1\le i\le \frac{mt-2}{2}$, $mt{K}_{16}$, and $\frac{mt}{2}{K}_2\left[16\right]$.

Let $0\le x\le \frac{mt-4}{2}$. We replace any $P_i$ $(1\le i\le x)$ with an HW$(C_m\left[16\right];16,m;16,0)$ and $P_i$ $(x+1\le i\le \frac{mt-4}{2})$ with an HW$(C_m\left[16\right];16,m;0,16)$ from Lemma 10.

Similarly to the above case, $P_{\frac{mt-2}{2}}\cup mt{K}_{16}$ can be partitioned into r $C_{16}$-factors, $23-r$ $C_m$-factors, and a one-factor on the whole vertex-set $Z_{mt}\times Z_{16}$ for odd $m\ge 9$ and $7\le r\le 23$. Furthermore, using Theorem 1, $\frac{mt}{2}{K}_2\left[16\right]$ can be decomposed into eight $C_{16}$-factors.

It is not difficult to calculate the number $\alpha$ of $C_{16}$-factors and the number $\beta$ of $C_m$-factors. We obtained $\alpha =16x+r+8$ and $\beta =16\times (\frac{mt-4}{2}-x)+23-r=8mt-1-(16x+r+8)$ for $0\le x\le \frac{mt-4}{2}$ and $7\le r\le 23$. We check that the range for $\alpha$ is 15 to $8mt-1$.  □

5. Concluding Remarks

We are working on the existence of an HW$(v;2^l,m;\alpha ,\beta )$ for an odd m. Theorem 4 completes the proof of the case $l=4$ and $\alpha \ge 15$. The construction method in this paper is still useful for other cases. We have some preliminary results and believe that the following conjecture is valid, but there is still a long way to go before the whole problem can be solved completely.

Conjecture 1.

For any odd $m\ge 2^{l-1}+1$, $(\alpha ,\beta )\in$ HWP$(2^{l}mt;2^l,m)$ if and only if $\alpha +\beta =2^{l-1}mt-1$, where $\alpha \ge 0$, $\beta \ge 0$, and $t\ge 1$, except possibly when $\alpha \in [1,2^{l-1}-2]$ and t is odd or $\alpha \in [1,2^{l-1}-2]\cup [2^{l-1},2^l-2]$ and t is even.