Uniformly Resolvable Decompositions of K_v-I into n-Cycles and n-Stars, for Even n

Abstract

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set $\Gamma$ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph $X\in \Gamma$. In this article we completely solve the existence problem for decompositions of $K_v$-I into $C_n$-factors and $K_{1,n}$-factors in the case when n is even.

1. Introduction and Definitions

For any graph G, let $V\left(G\right)$ and $E\left(G\right)$ be the vertex-set and the edge-set of G, respectively. Throughout the paper $K_v$ will denote the complete graph on v vertices, while $K_v\backslash K_h$ will denote the graph with $V\left(K_v\right)$ as vertex-set and $E\left(K_v\right)\backslash E\left(K_h\right)$ as edge-set (this graph is sometimes referred to as a complete graph of order v with a hole of size h).

Given a set $\Gamma$ of pairwise non-isomorphic graphs, a Γ-decomposition (or Γ-design)of a graph G is a decomposition of the edge-set of G into subgraphs (called blocks) that are isomorphic to some element of $\Gamma$. A $\Gamma$-factor of G is a spanning subgraph of G whose components are isomorphic to a member of $\Gamma$. If $X\in \Gamma$, then an X-factor is a spanning subgraph whose components are isomorphic to X. A $\Gamma$-decomposition of G is resolvable if its blocks can be partitioned into $\Gamma$-factors and is called a Γ-factorization of G. A $\Gamma$-factorization of G is called uniform if each factor is an X-factor for some graph $X\in \Gamma$. A $K_2$-factorization of G is known as a 1-factorization and its factors are called 1-factors; it is well known that a 1-factorization of $K_v$ exists if and only if v is even ([1]). A $C_k$-factorization of $K_v$ exists if and only if $3\le k\le v$, v and k are odd, and $v\equiv 0(modk)$ ([2]).

A $\Gamma$-isofactorization of G is a $\Gamma$-factorization with isomorphic factors. If $\Gamma$ is the set of all possible cycles of $K_v$, then determining the existence of possible $\Gamma$-isofactorizations of $K_v$ with an odd v is known as the Oberwolfach Problem. It was first posed in 1967 by Gerhard Ringel and asks whether it is possible to seat an odd number v of mathematicians at n round tables in $(v-1)/2$ meals so that each mathematician sits next to everyone else exactly once. If the n round tables are of sizes $p_1,p_2,\dots ,p_n$ (with $p_1+p_2+\cdots +p_n=v$), the Oberwolfach Problem asks for an isofactorization of $K_v$ with factors whose components are isomorphic to cycles of length $p_1,p_2,\dots ,p_n$. It is easy to see that such a factorization can exist only if v is odd. For even v, it is common to instead decompose $K_v$-I, with the complete graph with the edges of a 1-factor removed. The uniform Oberwolfach problem (all cycles of a factor have the same size) has been completely solved by Alspach and Häggkvist [3] and Alspach, Schellenberg, Stinson and Wagner [2] .

Additional existence problems for $\Gamma$-factorizations of $K_v$ or $K_v$-I have been studied and many results have been obtained, especially on uniformly resolvable $\Gamma$-decompositions: when $\Gamma$ is a set of two complete graphs of an order of at most five in [4,5,6,7]; when $\Gamma$ is a set of two or three paths on two, three or four vertices in [8,9,10]; for $\Gamma =\{P_3,K_3+e\}$ in [11]; for $\Gamma =\{K_3,K_{1,3}\}$ in [12]; for $\Gamma =\{C_4,P_3\}$ in [13]; for $\Gamma =\{K_3,P_3\}$ in [14]; for $\Gamma =\{K_2,K_{1,3}\}$ in [15,16]; for $\Gamma =\{K_2,K_{1,4}\}$ in [17]. Most famous is the variation of the Oberwolfach problem known as the Hamilton-Waterloo problem. In this problem the meals for the dining mathematicians take place at two different venues. Hence a decomposition of $K_v$ or $K_v$-I is sought where the factors can be of either one of two types. In particular, the uniform case asks for a decomposition of $K_v$ or $K_v$-I into $C_p$-factors and $C_q$-factors. Thus the round tables in one venue sit p mathematicians, whereas the tables in the other venue each sit q. Of course, in this case p and q must divide v and $\Gamma =\{C_p,C_q\}$.

A uniformly resolvable $\{X,Y\}$-decomposition of G into exactly r X-factors and s Y-factors is abbreviated as $(X,Y)$-URD$(G;r,s)$. If $G=K_v$ we simply write $(X,Y)$-URD$(v;r,s)$. In this paper, we study uniformly resolvable $\Gamma$-decompositions in the case when $\Gamma =\{C_n,K_{1,n}\}$. The existence problem of a $(C_n,K_{1,n})$-URD$(v;r,s)$ was solved for $n=2$ ([9], note that $C_2=K_2$) and $n=3$ ([12]). Here we deal with the case when n is even and greater or equal to 4. For an even n, it is known that a $(C_n,K_{1,n})$-URD$(v;0,s)$ exists if and only if $v\equiv 1(mod2n)$ and $v\equiv 0(modn+1)$ ([18]), while, when v is even, no $(C_n,K_{1,n})$-URD$(v;r,s)$ exists with $r>0$ because otherwise, $2(n+1)r+2ns=(n+1)(v-1)$, which is clearly impossible. Hence we study the existence problem for $(C_n,K_{1,n})$-URD$(K_v-I;r,s)$, which is denoted by $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ and, since n and $n+1$ must divide v, we assume that $v\equiv 0(modn(n+1\left)\right)$. Furthermore, since $\frac{(v-2)(n+1)}{2n}\notin \mathbb{N}$, necessarily $r>0$.

For $v\equiv 0(modn(n+1\left)\right)$, defined the set $J\left(v\right)$ according to the following Table 1.

Table 1. The set $J\left(v\right)$.

v	$\mathit{J}\left(\mathit{v}\right)$
$0(mod2n(n+1\left)\right)$	$(\frac{v-2}{2}-nx,(n+1)x),x=0,1,\dots ,\frac{v-2n}{2n}$
$n(n+1)(mod2n(n+1\left)\right)$	$(\frac{v-2}{2}-nx,(n+1)x),x=0,1,\dots ,\frac{v-n}{2n}$

We completely solve the existence problem of a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ by proving the following result.

Theorem 1.

Let $v\equiv 0(modn(n+1\left)\right)$. There exists a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ if and only if $(r,s)\in J\left(v\right)$.

2. General Constructions and Related Structures

A $\Gamma$-decomposition of $K_{u\left(g\right)}$, the complete multipartite graph with u parts of size g, is known as a group divisible decomposition ($\Gamma$-GDD for short) of type $g^u$; the parts of size g are called the groups. (If $\Gamma$ consists of complete subgraphs, then a $\mathsf{GDD}$ is called a group divisible design). When $\Gamma =\left\{G\right\}$, we simply write G-GDD, and when $G=K_n$, we refer to such a group divisible design as an n-GDD. We denote a (uniformly) resolvable $\Gamma$-GDD by $\Gamma$-(U)RGDD. Specifically, an $(X,Y)$-URGDD with r X-factors and s Y-factors is denoted by $(X,Y)$-URGDD$(r,s)$. It is easy to deduce that the number of G-factors of a G-RGDD is $\alpha =\frac{g(u-1)\left|V\right(G\left)\right|}{2\left|E\right(G\left)\right|}$.

If the blocks of a $\Gamma$-GDD of type $g^u$ can be partitioned into partial factors, each of which contains all vertices except those of one group, we refer to such a decomposition as a Γ-frame (an n-frame if $\Gamma =\left\{K_n\right\}$). For a fixed positive integer d, if $\Gamma$ is a set of d-regular graphs, then it is easy to deduce that the number of partial factors missing a specified group is $\alpha =\frac{g}{d}$.

A $\Gamma$-decomposition of $K_{v+h}\backslash K_h$ is known as an incomplete Γ-design of order $v+h$ with a hole of size h. We are interested in incomplete resolvable $\Gamma$-designs, which will be used in the “filling” and “frame”-constructions of this section. These designs have two types of factors: partial factors, which cover every vertex except the ones in the hole; and full factors, which cover every vertex of $K_{v+h}$.

Specifically, a $(X,Y)$-IURD$(v+h,h;[r^{\prime },s^{\prime }],[r,s])$ is a uniformly resolvable $(X,Y)$-decomposition of $K_{v+h}\backslash K_h$ with $r^{\prime }$ partial X-factors and $s^{\prime }$ partial Y-factors that cover every vertex not in the hole, and r X-factors and s Y-factors that cover every vertex of $K_{v+h}$.

Given a graph G and a positive integer t, $G_{\left(t\right)}$ will denote the graph on $V\left(G\right)\times {\mathbb{Z}}_t$ with edge-set $\{\{x_i,y_j\}:\{x,y\}\in E\left(G\right),i,j\in {\mathbb{Z}}_t\}$, where the subscript notation $a_i$ is used to denote the pair $(a,i)$. The graph $G_{\left(t\right)}$ is said to be obtained from G by expanding each vertex t times. When $G=K_n$, the graph $G_{\left(t\right)}$ is the complete equipartite graph $K_{\underset{n\mathrm{times}}{\underbrace{t,t,\dots ,t}}}$ with n parts of size t and will be denoted by $K_{n\left(t\right)}$; while $C_{n\left(t\right)}$ will denote the graph $G_{\left(t\right)}$ where G is an n-cycle.

Remark 1.

Note that the graph $G_{\left(t\right)}$ admits t 1-factors corresponding to each 1-factor of G; for instance, starting from the two 1-factors of a $2m$-cycle, 2t 1-factors of $C_{2m\left(t\right)}$ can be obtained (t 1-factors for each 1-factor of the $2m$-cycle).

For any two pairs of non-negative integers $(r,s)$ and $(r^{\prime },s^{\prime })$, define $(r,s)+(r^{\prime },s^{\prime })=(r+r^{\prime },s+s^{\prime })$. If X and $X^{\prime }$ are two sets of pairs of non-negative integers and a is a positive integer, then $X+X^{\prime }$ will denote the set $\{(r,s)+(r^{\prime },s^{\prime }):(r,s)\in X,(r^{\prime },s^{\prime })\in X^{\prime }\}$ and $a\ast X$ will denote the set of all pairs of non-negative integers that can be obtained by adding any a pairs of X together (repetitions of elements of X are allowed).

Construction 1.

(GDD-Construction) Let $\mathcal{G}$ be a Γ-RGDD of type $g^u$, where Γ is a set of graphs of order $n\ge 2$, and let t be a positive integer. If for any fixed factor $F_i$, $i=1,2,\dots ,\alpha$, there exists an $(X,Y)$-URD$(\overline{r},\overline{s})$ of $B_{\left(t\right)}$ for each $B\in F_i$ and for each $(\overline{r},\overline{s})\in J_i$, then so does an $(X,Y)$-URGDD$(r,s)$ of type ${\left(gt\right)}^u$ for each $(r,s)\in J_1+J_2+\cdots +J_{\alpha }$.

Proof. 

Expand each vertex t times. For $i=1,2,\dots ,\alpha$, for each block B of $F_i$ on $V\left(B\right)\times {\mathbb{Z}}_t$ place a copy of an $(X,Y)$-URD$(r_i,s_i)$ of $B_{\left(t\right)}$ with $(r_i,s_i)\in J_i$. Thus we obtain an $(X,Y)$-URGDD$(r,s)$ of type ${\left(gt\right)}^u$ with $r={\sum }_{i=1}^{\alpha }{r}_i$ and $s={\sum }_{i=1}^{\alpha }{s}_i$, and so $(r,s)\in J_1+J_2+\cdots +J_{\alpha }$. □

Construction 2.

(Filling Construction) Suppose there exists a $(X,Y)$-URGDD$(r,s)$ of type $g^u$ for each $(r,s)\in J$. If there exists an $(X,Y)$-URD$(g;r^{\prime },s^{\prime })$, for each $(r^{\prime },s^{\prime })\in J^{\prime }$, then so does:

(i) 

an $(X,Y)$-IURD$(ug,g;[r^{\prime },s^{\prime }],[r,s])$ for each $(r^{\prime },s^{\prime })$ $\in J^{\prime }$ and $(r,s)\in J$;

(ii) 

an $(X,Y)$-URD$(ug;\overline{r},\overline{s})$, for each $(\overline{r},\overline{s})\in J^{\prime }+J$.

Proof. 

Fix any pairs $(r,s)\in J$ and $(r^{\prime },s^{\prime })\in J^{\prime }$, and start with an $(X,Y)$-URGDD$(r,s)$ with u groups of size g, $G_i$, $i=1,2,\dots ,u$. For every $i=2,3,\dots ,u$, place a copy of an $(X,Y)$-URD$(g;r^{\prime },s^{\prime })$ on $G_i$ to obtain an $(X,Y)$-IURD$(gu,g;[r^{\prime },s^{\prime }],[r,s])$ with $G_1$ as the hole. Finally, on $G_1$ place a copy of an $(X,Y)$-URD$(g;r^{\prime },s^{\prime })$ to obtain an $(X,Y)$-URD$(gu;r^{\prime }+r,s^{\prime }+s)$. □

Remark 2.

Note that the “filling” technique allows us to construct an $(X,Y)$-URD$(v+h;r^{\prime }+r,s^{\prime }+s)$ whenever an $(X,Y)$-IURD$(v+h,h;[r^{\prime },s^{\prime }],[r,s])$ and an $(X,Y)$-URD$(h;r^{\prime },s^{\prime })$ are given.

Construction 3.

(Frame-Construction) Let $\mathcal{F}$ be a Γ-frame of type $g^u$, where Γ is a set of graphs of order $n\ge 2$ and the number of partial factors missing any fixed group is α, and let t, h and v be positive integers such that $v=gtu+h$. If there exists:

(i) 

An $(X,Y)$-URD$(\overline{r},\overline{s})$ of $G_{\left(t\right)}$ for each $G\in \Gamma$ and for each $(\overline{r},\overline{s})\in J$;

(ii) 

An $(X,Y)$-IURD$(gt+h,h;[r^{\prime },s^{\prime }],[\overline{\overline{r}},\overline{\overline{s}}])$ for each $(r^{\prime },s^{\prime })$ $\in J^{\prime }$ and $(\overline{\overline{r}},\overline{\overline{s}})\in \alpha \ast J$;

(iii) 

An $(X,Y)$-URD$(h;r^{\prime },s^{\prime })$ for each $(r^{\prime },s^{\prime })\in J^{\prime }$;

then so does an $(X,Y)$-URD$(v;r,s)$ for each $(r,s)\in J^{\prime }+u\alpha \ast J$ exist.

Proof. 

Let $A_i$, $i=1,2,\dots ,u$, be the groups of $\mathcal{F}$ and for $j=1,2,\dots ,\alpha$, let $F_{ij}$ be the j-th partial factor that misses the group $A_i$. Expand each vertex t times and add a set H of t extra vertices. For $j=1,2,\dots ,\alpha$, let $F_{ij}$ be the j-th partial factor that misses the group $G_i$. For each block $B\in F_{ij}$, on $v\left(B\right)\times {\mathbb{Z}}_t$ place a copy, ${\mathcal{D}}_{ij}\left(B\right)$, of an $(X,Y)$-URD$(r_{ij},s_{ij})$ of $B_{\left(t\right)}$ with $(r_{ij},s_{ij})\in J$. For $i=1,2,\dots ,u$, on $H\cup (A_i\times {\mathbb{Z}}_t)$ place a copy ${\mathcal{D}}_i$ of an $(X,Y)$-IURD$(gt+h,h;[r^{\prime },s^{\prime }],[r_i,s_i])$ with $(r^{\prime },s^{\prime })\in J^{\prime }$ and $(r_i,s_i)={\sum }_{j=1}^{\alpha }\left(r_{ij},s_{ij}\right)\in \alpha \ast J$. For every $i=1,2,\dots ,u$, combine all of the factors of ${\mathcal{D}}_{ij}\left(B\right)$, $B\in F_{ij}$, along with the full factors of ${\mathcal{D}}_i$ to obtain $\overline{r}$X-factors and $\overline{s}$Y-factors, where $(\overline{r},\overline{s})={\sum }_{i=1}^u(r_i,s_i)\in u\alpha \ast J$. Now, fill the hole H with a copy $\mathcal{D}$ of an $(X,Y)$-URD$(h;r^{\prime },s^{\prime })$ with $(r^{\prime },s^{\prime })\in J^{\prime }$. Combine the factors of $\mathcal{D}$ with the partial factors of ${\mathcal{D}}_i$ to obtain further $r^{\prime }$X-factors and $s^{\prime }$Y-factors with $(r^{\prime },s^{\prime })\in J^{\prime }$. The result is an $(X,Y)$-URD$(v;r,s)$ where $(r,s)=(r^{\prime }+\overline{r},s^{\prime }+\overline{s})\in J^{\prime }+u\alpha \ast J$. □

We quote the following known results for a later use.

Lemma 1

(Ref. [19]). For $l\ge 3$ and $u\ge 2$, there exists a $C_l$-RGDDof type $g^u$ if and only if $g(u-1)\equiv 0(mod2)$, $gu\equiv 0(modl)$, $l\equiv 0(mod2)$ if $u=2$, and $(g,u,l)\notin \left\{\right(2,3,3),(6,3,3),(2,6,3),(6,2,6\left)\right\}$.

Lemma 2

(Ref. [20]). A $\{C_3,C_4\}$-frame of type $g^u$ exists if and only if $u\ge 3$ and $g\equiv 0(mod2)$.

3. Necessary Conditions and Preliminary Lemmas

Let $n\equiv 0(mod2)$, $n\ge 4$. To start with, in this section we will give necessary conditions for the existence of a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ and then we will prove some basic lemmas that are useful for obtaining our main result. Let $p=n(n+1)$.

Lemma 3.

Let $v\equiv 0(modp)$. If there exists a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ then $(r,s)\in J\left(v\right)$.

Proof. 

By the resolvability:

$$\frac{rnv}{n}+\frac{nsv}{n+1}=\frac{v(v-2)}{2},$$

and hence

$$2(n+1)r+2ns=(n+1)(v-2).$$

(1)

Denote by R the set of r $C_n$-factors and by S the set of s $K_{1,n}$-factors. Since the factors of R are regular of degree 2, every vertex of $K_v$-I is incident to r $C_n$-factors in R and $(v-2)-2r$ edges in S. Assume that any fixed vertex appears in x factors of S with degree n and in y factors of S with degree 1. Since

$$x+y=sandnx+y=v-2-2r,$$

equality (1) gives us:

$$(n+1)(v-2-nx-y)+2n(x+y)=(n+1)(v-2),$$

which implies $y=nx$ and $s=(n+1)x$. Replacing $s=(n+1)x$ in Equation (1) provides $r=\frac{v-2}{2}-nx$, where $x<\frac{v-2}{2n}$ (because r is a positive integer) and so $0\le x\le \lfloor \frac{v-2}{2n}\rfloor$. □

In what follows, we will denote by $(a_1,a_2,\dots ,a_n)$ the n-cycle on $\{a_1,a_2,\dots ,a_n\}$ with edge-set $\{\{a_1,a_2\},$$\{a_2,a_3\},\dots ,\{a_{n-1},a_n\},\{a_n,a_1\}\}$, and by $(a;a_1,a_2,\dots ,a_n)$ the graph $K_{1,n}$ on the vertex-set $\{a,a_1,a_2,\dots ,a_n\}$ with edge-set $\{\{a,a_1\},\{\{a,a_2\},\dots ,\{a,a_n\}\}$. If G is a graph whose vertices belong to ${\mathbb{Z}}_v$, then we call orbit of B under $Z_v$ the set $\left(G\right)=\{G+i:i\in {\mathbb{Z}}_v\}$, where $G+i$ is the graph with $V(G+i)=\{a+i:a\in V(G\left)\right\}$ and $E(G+i)=\left\{\right\{a+i,b+i\}:\{a,b\}\in V(G\left)\right\}$.

Lemma 4.

A $(C_n,K_{1,n})$-URD$(r,s)$ of $C_{n\left(t\right)}$ where $t=n+1$ exists for $(r,s)=(n+1,0),(1,n+1)$.

Proof. 

Start from the cycle $C=(0,1,\dots ,n-1)$ on ${\mathbb{Z}}_n$ and expand it $t=n+1$ times. For the case $(r,s)=(1,n+1)$, take the following factors:

$F=\{(0_j,1_j,\dots ,{(n-1)}_j):j=0,1,\dots ,n\}$,

$F_j=\{(i_j;{(1+i)}_{j+1},{(1+i)}_{j+2},\dots ,{(1+i)}_{j+n}):i\in {\mathbb{Z}}_n\}$, $j\in {\mathbb{Z}}_{n+1}$.

For the case $(r,s)=(n+1,0)$, take the following $C_n$-factors:

$F_j^{\prime }=\{(0_i,1_{i+j},2_i,3_{i+j},\dots ,{(n-2)}_i,{(n-1)}_{i+j}):i\in {\mathbb{Z}}_{n+1}\}$, $j\in {\mathbb{Z}}_{n+1}$. □

Lemma 5.

A G-factorization of $G_{\left(n\right)}$ exists for $G=C_n,K_{1,n}$.

Proof. 

For $G=C_n$, start from the n-cycle $(1,2,\dots ,n)$ and on $\{1,2,\dots ,n\}\times {\mathbb{Z}}_n$ consider the following $C_n$-factors:

$F_i=\{(1_i,2_{i+j},3_i,4_{i+j},\dots ,n_{i+j}):j\in {\mathbb{Z}}_n\}$, $i\in {\mathbb{Z}}_n$.

For $G=K_{1,n}$, start from $(0;1,2,\dots ,n)$ and on $\{0,1,2,\dots ,n\}\times {\mathbb{Z}}_n$ consider the following $K_{1,n}$-factors:

$F_i^{\prime }=\{(0_i;1_0,1_1,\dots ,1_{n-1}),(0_{i+1};2_0,2_1,\dots ,2_{n-1}),\dots ,(0_{i-1};n_0,n_1,\dots ,n_{n-1})\}$, $i\in {\mathbb{Z}}_n$. □

Lemma 6.

There exists a $(C_n,K_{1,n})$-URD$(n,n(n+1\left)\right)$ of $C_{n\left(p\right)}$.

Proof. 

Let $F_i$, $i=1,2,\dots ,n$ be the $C_n$-factorization of $C_{n\left(n\right)}$ given by Lemma 5. Expand each vertex $t=n+1$ times. For $i=1,2,\dots ,n$, for each n-cycle C of $F_i$ on $V\left(C\right)\times {\mathbb{Z}}_t$ place a copy of a $(C_n,K_{1,n})$-URD$(1,n+1)$ of $C_{n\left(t\right)}$ (given by Lemma 4) to get a $(C_n,K_{1,n})$-URD$(n,n(n+1\left)\right)$ of $C_{n\left(p\right)}$. □

It is not difficult to generalize Lemma 4.8 of [17] so as to obtain a more general result that holds for any even n.

Lemma 7.

A $(C_n,K_{1,n})$-URD$(0,{(n+1)}^2)$ of $C_{m\left(p\right)}$ exists for every $m\ge 3$.

Lemma 8.

There exists a $(C_n,K_{1,n})$-URD${}^{*}(2(n+1);0,n+1)$.

Proof. 

The orbit of $B=(0;1,2,\dots ,n)$ under ${\mathbb{Z}}_{2(n+1)}$ is the block set of a $K_{1,n}$-decomposition of $K_{2(n+1)}-I$ and can be partitioned into the $n+1$ factors $F_i=\{B+i+(n+1)j,j=0,1\}$, for $i=0,1,\dots ,n$, to obtain the required design. □

Lemma 9.

Let $v=pk$, $k\ge 1$. A $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ exists for every $(r,s)\in \left\{(\frac{v-2}{2}-nx,\right.$ $\left.(n+1)x):x=0,1,\dots ,\frac{kn}{2}\right\}$.

Proof. 

Start from a $C_n$-RGDD of type $2^{\frac{nk}{2}}$, which exists by Lemma 1 and has $\alpha =\frac{nk}{2}-1$ factors. Applying the GDD-construction with $t=n+1$ gives a $(C_n,K_{1,n})$-URGDD$(\overline{r},\overline{s})$ of type ${\left[2(n+1)\right]}^{\frac{nk}{2}}$ for each $(\overline{r},\overline{s})\in (\frac{nk}{2}-1)*\{(n+1,0),(1,n+1)\}$ (the input designs are given by Lemma 4). Now fill the groups with copies of a $(C_n,K_{1,n})$-URD${}^{*}(2(n+1);0,n+1)$ from Lemma 8 to get a $(C_n,K_{1,n})$-URD${}^{*}(pk;r,s)$ for each $(r,s)\in \left\{(0,n+1)\right\}+(\frac{nk}{2}-1)\ast \{(n+1,0),(1,n+1)\}=\left\{\frac{v-2}{2}-nx,(n+1)x):x=1,\dots ,\frac{nk}{2}\right\}$. The missing case ($x=0$) corresponds to a $C_n$-factorization of $K_{pk}-I$, which is known to exist (see [21]). □

Lemma 10.

A $(C_n,K_{1,n})$-URD${}^{*}(p;r,s)$ exists for every $(r,s)\in J\left(p\right)$.

Proof. 

It follows by Lemma 9 for $k=1$. □

Lemma 11.

A $(C_n,K_{1,n})$-URD${}^{*}(2p;r,s)$ exists for every $(r,s)\in J\left(2p\right)$.

Proof. 

It follows by Lemma 9 for $k=2$. □

Lemma 12.

A $(C_n,K_{1,n})$-URGDD$(r,s)$ of type $p^{1+2k}$, $k\ge 1$, exists for every $(r,s)\in \{kp-nx,(n+1\left)x\right):$ $x=0,1,\dots ,kn,k(n+1)\}$.

Proof. 

Applying the GDD-construction with $t=n+1$ to a $C_n$-RGDD of type $n^{1+2k}$ (which exists by Lemma 1 and has $\alpha =nk$ factors) gives a $(C_n,K_{1,n})$-URGDD$(\overline{r},\overline{s})$ of type $p^{1+2k}$ for each $(\overline{r},\overline{s})\in nk*\{(n+1,0),(1,n+1)\}=\{pk-nx,(n+1)x):x=0,1,\dots ,nk\}$ (the input designs are given by Lemmas 4). For $(r,s)=(0,k{(n+1)}^2)$, apply the GDD-construction with $t=p$ to a $C_{1+2k}$-RGDD of type $1^{1+2k}$, which exists by Lemma 1 and has $\alpha =k$ factors (the input designs are given by Lemma 7). □

Lemma 13.

Let $v=p+2pk$, $k>0$. A $(C_n,K_{1,n})$-IURD${}^{*}(p+2pk,p;[r^{\prime },s^{\prime }],[r,s])$ exists for each $(r^{\prime },s^{\prime })$ $\in J\left(p\right)$ and $(r,s)\in \{pk-nx,(n+1\left)x\right):$ $x=0,1,\dots ,nk,(n+1)k\}$. In addition, if $k\le \frac{n}{2}+1$, then a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ exists for every $(r,s)\in J\left(v\right)$.

Proof. 

It follows by applying the filling construction to the GDD from Lemma 12 and using copies of a $(C_n,K_{1,n})$-URD${}^{*}(p;r,s)$ from Lemma 10 as input designs. □

As a consequence of the previous lemma we have the following two lemmas.

Lemma 14.

A $(C_n,K_{1,n})$-IURD$(3p,p;[r^{\prime },s^{\prime }],[r,s])$ exists for each $(r^{\prime },s^{\prime })$ $\in J\left(p\right)$ and $(r,s)\in \left\{\right(p-nx,(n+1)x),x=0,1,\dots ,n+1\}$.

Lemma 15.

A $(C_n,K_{1,n})$-URD${}^{*}(3p;r,s)$ exists for every $(r,s)\in J\left(3p\right)$.

Lemma 16.

A $(C_n,K_{1,n})$-URGDD$(r,s)$ of type ${\left(2p\right)}^k$, $k\ge 2$, exists for every $(r,s)\in \left\{\right((k-1)p-nx,(n+1)x):$ $x=0,1,\dots ,n(k-1),(n+1)(k-1)\}$.

Proof. 

Applying the GDD-construction with $t=n+1$ to a $C_n$-RGDD of type ${\left(2n\right)}^k$, $k\ge 2$, (which exists by Lemma 1 and has $\alpha =n(k-1)$ factors) gives a $(C_n,K_{1,n})$-URGDD$(\overline{r},\overline{s})$ of type ${\left(2p\right)}^k$ for each $(\overline{r},\overline{s})\in (k-1)n*\{(n+1,0),(1,n+1)\}=\{p(k-1)-nx,(n+1)x):$ $x=0,1,\dots ,n(k-1)$ (the input designs are given by Lemmas 4). For $(r,s)=(0,(k-1){(n+1)}^2)$, apply the GDD-construction with $t=p$ to a $C_{2k}$-RGDD of type $2^k$, $k\ge 2$, which exists by Lemma 1 and has $\alpha =k-1$ factors (the input designs are given by Lemma 7). □

Lemma 17.

A $(C_n,K_{1,n})$-URGDD$(r,s)$ of type ${\left(2p\right)}^2$ exists for every $(r,s)\in \left\{\right(p-nx,(n+1)x):x=0,1,\dots ,n+1\}$.

Proof. 

It follows by Lemma 16 for $k=2$. □

Lemma 18.

A $(C_n,K_{1,n})$-URGDD$(r,s)$ of $C_{4\left(p\right)}$ exists for every $(r,s)\in \{p-nx,(n+1\left)x\right):x=0,1,\dots ,n+1\}$.

Proof. 

It follows by Lemma 17 because the graph $K_{2p,2p}$ is isomorphic to $C_{4\left(p\right)}$. □

Lemma 19.

A $(C_n,K_{1,n})$-URGDD$(r,s)$, $n\ne 6$, of type $p^2$ exists for every $(r,s)\in \{\frac{p}{2}-nx,(n+1)x):$ $x=0,1,\dots ,\frac{n}{2}\}$.

Proof. 

For $n\ne 6$, applying the GDD-construction with $t=n+1$ to a $C_n$-RGDD of type $n^2$ (which exists by Lemma 1 and has $\alpha =\frac{n}{2}$ factors) gives a $(C_n,K_{1,n})$-URGDD$(\overline{r},\overline{s})$ of type $p^2$ for each $(\overline{r},\overline{s})\in \frac{n}{2}*\{(n+1,0),(1,n+1)\}=\left\{\frac{p}{2}-nx,(n+1)x):x=0,1,\dots ,\frac{n}{2}\right\}$ (the input designs are given by Lemmas 4). □

Lemma 20.

A $(C_n,K_{1,n})$-URGDD$(r,s)$ of type ${\left(2p\right)}^3$ exists for every $(r,s)\in \left\{\right(2p-nx,(n+1)x):x=0,1,\dots ,2(n+1\left)\right\}$.

Proof. 

By Lemma 16 a $(C_n,K_{1,n})$-URGDD$(r,s)$ of type ${\left(2p\right)}^3$ exists for every $(r,s)\in \left\{\right(2p-nx,(n+1)x):x=0,1,\dots ,2n,2n+2\}$. We need to solve the case for $x=2n+1$. For $n\ne 6$, apply the GDD-construction with $t=p$ to a $(C_6,K_2)$-URGDD$(1,2)$ of type $2^3$ (which can be obtained from a $C_6$-RGDD of type $2^3$ by replacing one 6-cycle with two 1-factors) and get a $(C_n,K_{1,n})$-URGDD$(r,s)$ of type ${\left(2p\right)}^3$ with $(r,s)=2(\frac{n}{2},\frac{n}{2}(n+1))+(0,{(n+1)}^2)=(n,(n+1)(2n+1))$ (the input designs are two copies of a $(C_n,K_{1,n})$-URGDD$(\frac{n}{2},\frac{n}{2}(n+1))$ of type $p^2$ given by Lemma 19, and a copy of a $(C_n,K_{1,n})$-URD$(0,{(n+1)}^2)$ of $C_{6\left(p\right)}$ from Lemma 7). For $n=6$, apply the GDD-construction with $t=p=42$ to a $C_6$-RGDD of type $2^3$ and get a $(C_6,K_{1,6})$-URGDD$(6,91)$ of type ${84}^3$ (the input designs are given by Lemmas 6 and 7). □

By Lemmas 11 and 20, and the filling constructions the following two lemmas follow.

Lemma 21.

A $(C_n,K_{1,n})$-IURD$(6p,2p;[r^{\prime },s^{\prime }],[r,s])$ exists for each $(r^{\prime },s^{\prime })$ $\in J\left(2p\right)$ and $(r,s)\in \left\{\right(2p-nx,(n+1)x),x=0,1,\dots ,2(n+1\left)\right\}$.

Lemma 22.

A $(C_n,K_{1,n})$-URD${}^{*}(6p;r,s)$ exists for each $(r,s)\in J\left(6p\right)$.

Lemma 23.

A $(C_n,K_{1,n})$-URD${}^{*}(10p;r,s)$ exists for every $(r,s)\in J\left(10p\right)$.

Proof. 

Apply the filling construction to a $(C_n,K_{1,n})$-URGDD$(r,s)$ of type ${\left(2p\right)}^5$ with $(r,s)\in \left\{\right(4p-nx,(n+1)x):x=0,1,\dots ,4n,4(n+1\left)\right\}$ (given by Lemma 16 for $k=5$) by using copies of a $(C_n,K_{1,n})$-URD${}^{*}(2p;r,s)$ from Lemma 11 as input designs. □

4. The Main Result

Lemma 24.

Let $v\equiv 0(mod4p)$. Then a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ exists for every $(r,s)\in J\left(v\right)$.

Proof. 

Let $v=4pk$, $k\ge 1$. Applying the GDD-construction with $t=2p$ to a 2-RGDD of type $1^{2k}$ (i.e., a 1-factorization of $K_{2k}$, which is known to have $\alpha =2k-1$ 1-factors) gives a $(C_n,K_{1,n})$-URGDD$(\overline{r},\overline{s})$ of type ${\left(2p\right)}^{2k}$ for each $(\overline{r},\overline{s})\in (2k-1)*\{(p-nx,(n+1)x):x=0,1,\dots ,n+1\}$ (the input designs are given by Lemma 17). Now fill the groups with copies of a $(C_n,K_{1,n})$-URD${}^{*}(2p;r^{\prime },s^{\prime })$ with $(r^{\prime },s^{\prime })\in J\left(2p\right)$ (from Lemma 11) to get a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ for each $(r,s)\in J\left(p\right)+(2k-1)*\left\{\right(p-nx,(n+1)x):x=0,1,\dots ,n+1\}=J\left(4pk\right)$. □

Lemma 25.

Let $v\equiv 2p(mod4p)$. Then a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ exists for every $(r,s)\in J\left(v\right)$.

Proof. 

Let $v=2p+4pk$, $k\ge 0$. The cases $v=2p,6p$ and $10p$ follow by Lemmas 11, 22 and 23, respectively. For $k\ge 3$, applying the frame-construction with $t=2p$ and $h=2p$ to a 2-frame of type $2^k$ (see [22]) gives a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ for each $(r,s)\in J\left(2p\right)+2k\ast \left\{\right(p-nx,(n+1)x):x=0,1,\dots ,n+1\}=J(2p+4pk)$ (the input designs are given by Lemmas 11, 17 and 21). □

Lemma 26.

Let $v\equiv p(mod2p)$. Then a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ exists for every $(r,s)\in J\left(v\right)$.

Proof. 

Let $v=p+2pk$, $k\ge 0$. The cases $v=p,3p$ and $5p$ follow by Lemmas 10, 13 and 15, respectively. For $l\ge 3$, apply the frame-construction with $t=p$ and $h=p$ to a $\{C_3,C_4\}$-frame of type $2^l$, which is known to exist ([20]) and have $\alpha =1$ factor missing in any fixed group, and get a $(C_n,K_{1,n})$-URD$(v;r,s)$ for each $(r,s)\in J\left(p\right)+k*\left\{\right(p-mx,(m+1)x),x=0,1,\dots ,m+1\}=J(p+2pk)$ (the input designs are given by Lemmas 10, 12, 14 and 18). □

As a consequence of Lemmas 24–26, our main result immediately follows.

Theorem 2.

Let $v\equiv 0(modn(n+1\left)\right)$. There exists a $(C_n,K_{1,n})$-URD${}^{*}(v;r,s)$ if and only if $(r,s)\in J\left(v\right)$.