The Complexity of Perfect Packings in Dense Graphs

Abstract

Let H be a graph of order h and let G be a graph of order n such that $h\mid n$. A perfect H-packing in G is a collection of vertex disjoint copies of H in G that covers all vertices of G. Hell and Kirkpatrick showed that the decision problem whether a graph G has a perfect H-packing is NP-complete if and only if H has a component which contains at least 3 vertices.

We consider the decision problem of containment of a perfect H-packing in graphs G under the additional minimum degree condition. Our main result shows that given any $\gamma >0$ and any n-vertex graph G with minimum degree at least $(1-1/\chi _{cr}(H)+\gamma )n$, the problem of determining whether G has a perfect H-packing can be solved in polynomial time, where $\chi _{cr}(H)$ is the critical chromatic number of H. This answers a question of Yuster negatively. Moreover, a hardness result of Kühn and Osthus shows that our main result is essentially best possible and closes a long-standing hardness gap for all complete multi-partite graphs H whose second smallest color class has size at least 2.

Appendices

A Proof of Proposition 1

We need the following simple counting result, which, for example, follows from the result of Erdős [5] on supersaturation.

Proposition 2

Given $\gamma '> 0$, $\ell _1, \dots , \ell _ k\in \mathbb {N}$, there exists $\mu >0$ such that the following holds for sufficiently large n. Let T be an n-vertex graph with a vertex partition $V_1 \cup \dots \cup V_d$. Suppose $i_1, \dots , i_k\in [d]$ and T contains at least $\gamma ' {n}^{k}$ copies of $K_k$ with vertex set $\{ v_1, \dots , v_k \}$ such that $v_1\in V_{i_1}$, $\dots , v_k\in V_{i_k}$. Then T contains at least $\mu {n}^{\ell _1+\cdots + \ell _k}$ copies of $K_{\ell _1, \dots , \ell _ k}$ whose jth part is contained in $V_{i_j}$ for all $j\in [k]$.

Proof

(Proof of Proposition 1 ). Write $L=L_{\mathcal {P}}^{\mu }(G)$ and $Q=Q(\mathcal {P}, L)$. It suffices to show that for any element $\mathbf {v}\in L_{\max }^{d}$, there exists $\mathbf {v}'=(v_1',\dots , v_d')$ such that $-(m-1)\le v_i'\le m-1$ for all $i\in [d]$ and $\mathbf {v}- \mathbf {v}'\in L$ – since the number of such $\mathbf {v}'$ is at most $(2m-1)^d$. Recall that since H is unbalanced, there exists a k-coloring with color class sizes $a_1\le \cdots \le a_k$ and $a_1< a_k$. Let $a=a_{k} - a_{1}<m$.

Define a graph P on the vertex set [d] such that $(i, j)\in P$ if and only if $e(G[V_i, V_j])\ge \gamma n^2$. We claim that if i and j are connected in P, then $a(\mathbf {u}_i-\mathbf {u}_j)\in L$. Indeed, first assume that $(i,j)\in P$. For each edge uv in $(V_i, V_j)$, by

$$ \delta (G)\ge (1-1/\chi _{cr}(H)+\gamma ) n\ge \left( 1- \frac{m-1}{(k-1)m} +\gamma \right) n, $$

it is easy to see that uv is contained in at least $\frac{1}{m^{k-2}} \left( {\begin{array}{c}n\\ k-2\end{array}}\right)$ copies of $K_k$. So there are at least $\gamma n^2 \cdot \frac{1}{m^{k-2}} \left( {\begin{array}{c}n\\ k-2\end{array}}\right) /\left( {\begin{array}{c}k\\ 2\end{array}}\right)$ copies of $K_k$ in G intersecting both $V_i$ and $V_j$. By averaging, there exists a k-array $(i_1, \dots , i_k)$, $i_j\in [d]$ where $i_1=i$ and $i_k=j$ such that G contains at least

$$ \frac{1}{d^{k-2}}\gamma n^2 \cdot \frac{1}{m^{k-2}} \left( {\begin{array}{c}n\\ k-2\end{array}}\right) /\left( {\begin{array}{c}k\\ 2\end{array}}\right) \ge \frac{\gamma }{m^{k-2}d^{k-2}k!}n^{k} $$

copies of $K_k$ with vertex set $\{ v_1, \dots , v_k \}$ such that $v_1\in V_{i_1}$, $\dots , v_k\in V_{i_k}$. By applying Proposition 2 with $\ell _i=a_i$, $i\in [k]$, we get that there are at least $\mu n^{m}$ copies of $K_{a_1, \dots , a_ k}$ in G whose jth part is contained in $V_{i_j}$ for all $j\in [k]$. We apply Proposition 2 again, this time with $\ell _i=a_i$, for all $2\le i\le k-1$ and $\ell _{1}=a_{k}$, $\ell _{k}=a_{1}$ and thus conclude that there are at least $\mu n^{m}$ copies of $K_{a_k, a_2,\dots , a_{k-1}, a_1}$ (with $a_1$ and $a_k$ exchanged) in G whose jth part is contained in $V_{i_j}$ for all $j\in [k]$. Taking subtraction of index vectors of these two types of copies gives that $a(\mathbf {u}_i-\mathbf {u}_j)\in L$. Furthermore, note that if i and j are connected by a path in P, we can apply the argument above to every edge in the path and conclude that $a(\mathbf {u}_i-\mathbf {u}_j)\in L$, so the claim is proved.

Now we separate two cases.

Case 1: $k\ge 3$ . In this case, we first show that P is connected. Indeed, we prove that for any bipartition $A\cup B$ of [d], there exists $i\in A$ and $j\in B$ such that $(i,j)\in P$. Let $V_A=\bigcup _{i\in A} V_i$ and $V_B=\bigcup _{j\in B}V_j$. Without loss of generality, assume that $|V_A|\le n/2$. Since $\delta (G)\ge \frac{1+(k-2)m}{(k-1)m} n \ge (1/2 + 1/(2m))n$, the number of edges in G that are incident to $V_A$ is at least

$$ |V_A| \cdot \left( \frac{1}{2} + \frac{1}{2m} \right) n - \left( {\begin{array}{c}|V_A|\\ 2\end{array}}\right) \ge \left( {\begin{array}{c}|V_A|\\ 2\end{array}}\right) + \frac{n}{4m}|V_A| \ge \left( {\begin{array}{c}|V_A|\\ 2\end{array}}\right) + \gamma n^2 |A| |B|, $$

where the last inequality follows from $|A| |B|\le d^2/4$ and $|V_i|\ge n/m$, $i\in [d]$. By averaging, there exists $i\in A$ and $j\in B$ such that $e(G[V_i, V_j])\ge \gamma n^2$ and thus $(i,j)\in P$.

Now let $\mathbf {v}=(v_1,\dots , v_d)\in L_{\max }^{d}$. We fix an arbitrary m-vector $\mathbf {w}\in L$ and let $\mathbf {v}_1 = \mathbf {v}- ( |\mathbf {v}|/m ) \mathbf {w}$ and thus $|\mathbf {v}_1| =0$. Since P is connected, the claim above implies that for any $i,j\in [d]$, $a(\mathbf {u}_i-\mathbf {u}_j)\in L$. Thus, we obtain the desired vector $\mathbf {v}'$ by making the difference of any two digits at most a. Since $|\mathbf {v}'|= |\mathbf {v}_1| =0$, we know that $-(m-1)\le v_i'\le m-1$ for all $i\in [d]$ and we are done.

Case 2: $k = 2$ . In this case we cannot guarantee that P is connected (we may even have some isolated vertices). First let i be an isolated vertex in P. By the definition of P, we know that $e(G[V_i, V\setminus V_i]) \le (d-1)\gamma n^2$. Since $\delta (G)\ge n/m$,

$$ e(G[V_i]) \ge \frac{1}{2} (|V_i| n/m - (d-1)\gamma n^2) \ge \frac{1}{4m} |V_i|^2. $$

Applying Proposition 2 on $V_i$ shows that there are at least $\mu n^m$ copies of $K_{a_1, a_2}$ in $V_i$, i.e., $m\mathbf {u}_i\in L$. Second, if $(i,j)\in P$, then applying Proposition 2 on $[V_i, V_j]$ gives that $a_1\mathbf {u}_i + a_2\mathbf {u}_j\in L$. So in both cases, for any component C in P, there exists an m-vector $\mathbf {w}\in L$ such that $\mathbf {w}|_{[d]\setminus C}=\mathbf 0$.

Now let $\mathbf {v}=(v_1,\dots , v_d)\in L_{\max }^{d}$. Consider the connected components $C_1$, $C_2, \dots , C_q$ of P, for some $1\le q\le d$. By the conclusion in the last paragraph, there exists $\mathbf {v}_1\in \mathbb {Z}^d$ such that $\mathbf {v}- \mathbf {v}_1\in L$ and for each component $C_i$, $0\le |\mathbf {v}_1|_{C_i}|\le m-1$ (for each component C, we take out a multiple of the vector $\mathbf {w}$ given by the last paragraph). Next, within each nontrivial component $C_i$, we can use the claim to ‘balance’ the digits, as in Case 1. At last we obtain a vector $\mathbf {v}'$ with the desired property.

B Proof of Lemma 1

In this subsection we prove Lemma 1. We will build a partition $\mathcal {P}=\{V_1,\dots , V_d\}$ of V(G) for some $d\le m$ such that every $V_i$ is $(H,\beta , t)$-closed for some $\beta >0$ and integer $t\ge 1$ in polynomial time. For any $v\in V(G)$, let $\tilde{N}_{H,\beta , i}(v)$ be the set of vertices in V(G) that are $(H,\beta , i)$-reachable to v.

We need the following lemma [19, Lemma 4.2], which was originally stated for k-uniform hypergraphs. We remark that the current form can be easily derived by defining a k-uniform hypergraph $G'$ where each k-set forms a hyperedge if and only if it spans a $K_k$ in G. For any vertex $u\in V(G)$, let W(u) be the collection of $(k-1)$-cliques $S\subseteq N(u)$.

Lemma 3

[19]. Given H and $\gamma '>0$, there exists $\alpha >0$ such that the following holds for sufficiently large n. For any n-vertex graph G, two vertices $x, y \in V(G)$ are $(H,\alpha ,1)$-reachable if the number of $(k-1)$-sets $S\in W(x)\cap W(y)$ with $|N(S)|\ge \gamma ' n$ is at least $\gamma ' \left( {\begin{array}{c}n\\ k-1\end{array}}\right)$, where $N(S)=\bigcap _{v\in S}N(v)$.

Proposition 3

Suppose $0< 1/n \ll \alpha \ll \gamma \ll 1/m$ and let G be an n-vertex graph with $\delta (G)\ge (1-1/\chi _{cr}(H)+\gamma ) n$. Then for any $v\in V(G)$, $|\tilde{N}_{H,\alpha , 1}(v)| \ge (1/m+\gamma )n$.

Proof

First note that for each $(k-1)$-clique S, we have $|N(S)|\ge (1/m + k\gamma )n$. Then by Lemma 3, for any distinct $u, v\in V(G)$, $u\in \tilde{N}_{H,\alpha , 1}(v)$ if $|W(u)\cap W(v)|\ge \gamma ^2 \left( {\begin{array}{c}n\\ k-1\end{array}}\right)$. By double counting, we have

$$ \sum _{S\in W(v)} |N(S)| < |\tilde{N}_{H,\alpha , 1}(v)|\cdot |W(v)|+n\cdot \gamma ^2 \left( {\begin{array}{c}n\\ k-1\end{array}}\right) . $$

Note that any S in the above inequality is a $(k-1)$-clique, thus $|N(S)| \ge (1/m + k\gamma )n$. On the other hand, it is easy to see that $|W(v)|\ge \frac{1}{m^{k-1}}\left( {\begin{array}{c}n\\ k-1\end{array}}\right)$. By $\gamma \ll 1/m$, we have

$$ |\tilde{N}_{H,\alpha , 1}(v)|> (1/m + k\gamma )n - \frac{\gamma ^2 n^k}{|W(v)|}\ge (1/m + \gamma )n. $$

   $\square$

The following lemma will be used to find the partition $\mathcal {P}$ in Lemma 1. Its proof is almost identical to the one of [6, Lemma 3.8] and thus we omit it.

Lemma 4

Given $0<\alpha \ll \{1/c, \delta '\}$, there exists constant $\beta >0$ such that the following holds for all sufficiently large n. Assume an n-vertex graph T satisfies that $|\tilde{N}_{H,\alpha , 1}(v)| \ge \delta ' n$ for any $v\in V(T)$ and every set of $c+1$ vertices in V(T) contains two vertices that are $(H,\alpha , 1)$-reachable. Then we can find a partition $\mathcal {P}$ of V(T) into $V_1,\dots , V_d$ with $d\le \min \{c, 1/\delta ' \}$ such that for any $i\in [d]$, $|V_i|\ge (\delta ' - \alpha ) n$ and $V_i$ is $(H,\beta , 2^{c-1})$-closed in T, in time $O(n^{2^{c-1} m+1})$.

Now we are ready to prove Lemma 1.

Proof

(Proof of Lemma 1 ). Fix $\gamma >0$. Without loss of generality, we may assume that $\gamma \ll 1/m$. We apply Lemma 4 with $c=m^{k-1}$, $\delta '=1/m+\gamma$ and $\alpha \ll \gamma$ and get $\beta >0$. Suppose

$$ 1/n_0\ll \beta \ll \alpha \ll \gamma \ll \delta '. $$

Let G be a graph on $n\ge n_0$ vertices satisfying $\delta (G)\ge (1-1/\chi _{cr}(H)+\gamma ) n$. By Proposition 3, for any $v\in V(G)$, $|\tilde{N}_{H,\alpha , 1}(v)| \ge \delta ' n$. By the degree condition and Lemma 3, for distinct $u, v\in V(G)$, u and v are $(H,\alpha , 1)$-reachable if $|W(u)\cap W(v)|\ge \gamma ^2 \left( {\begin{array}{c}n\\ k-1\end{array}}\right)$. So any set of $c+1$ vertices in V(G) contains two vertices that are $(H,\alpha , 1)$-reachable because $|W(u)|\ge \frac{1}{c}\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right)$ for any $u\in V(G)$, and $(c+1)/c-1\ge \left( {\begin{array}{c}c+1\\ 2\end{array}}\right) \gamma ^2$. So we can apply Lemma 4 on G and get a partition $\mathcal {P}=\{V_1, \dots , V_d\}$ of V(G) in time $O(n^{2^{c-1}m+1})$. Note that $|V_i|\ge (\delta '-\alpha )n \ge n/m$ for all $i\in [d]$. Also $d\le 1/\delta '\le m$ and each $V_i$ is $(H,\beta , 2^{c-1})$-closed.