Extremal problems for a matching and any other graph

In this paper, let $K_{r},K_{s,t}$ and $S_{k}$ denote the complete graph on $r$ vertices, complete bipartite graph with two parts of size $s$ and $t$, a star on $k$ vertices, respectively. Let $M_{s}$ denote a matching on $s$ edges. We use $|G|$ to denote the number of vertices of $G$. For a family of graphs $\mathcal{F}$, a graph is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subgraph. Let $G(n,s,H)$ denote the graph obtained from the complete bipartite graph $K_{s,n-s}$ by embedding a maximum $H$-free graph into the part of size $s$. For a subset $U\in V(G)$, we use $G[U]$ and $G-U$ to denote the subgraph induced by $U$ and $V(G)-U$, respectively.

For an integer $r$ and a family $\mathcal{F}$, the generalized Turán number ${\rm ex}(n,K_{r},\mathcal{F})$ is the maximum number of copies of $K_{r}$ in an $n$-vertex $\mathcal{F}$-free graph. Note that ${\rm ex}(n,K_{2},\mathcal{F})={\rm ex}(n,\mathcal{F})$, i.e., the classical Turán number. The generalized Turán number was firstly proposed by Alon and Shikhelman [3] in 2016. It has received a lot of attention in the past few years. Many classical results on Turán problem have been extended to generalized Turán number and some other interesting problem are studied too, see [7, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21].

In this paper, we mainly focus on the Turán problem concerning matching. The first result in this issue dates back to Erdős and Gallai [9], they proved

$${\rm ex}(n,M_{s+1})=\max\{e(G(n,s,K_{s+1})),e(K_{2s+1})\},$$

and determined the extremal graphs. This result was extended to generalized Turán number ${\rm ex}(n,K_{r},M_{s+1})$ by Wang [19]. Beyond that, Chvátal and Hanson [8], and independently by Balachandran and Khare [4] using different method, determined the value of ${\rm ex}(n,\{M_{s+1},S_{k+1}\})$ (The case for $s=k$ was proved early by Abbott, Hanson and Sauer [1]). Recently, Alon and Frankl [2] suggested to study ${\rm ex}(n,\{M_{s+1},F\})$ for any $F$. If there is an edge $e$ in $F$ such that $\chi(F-e)<\chi(F)$, then we call $F$ a color critical graph. They obtained the following results.

Follow these results, Gerbner [11] constructed some possible lower bounds of ${\rm ex}(n,\{M_{s+1},F\})$ and determined ${\rm ex}(n,\{M_{s+1},F\})$ apart from a constant additive term for some special bipartite graph $F$.

It appears likely that for $r\geq 3$, the function ${\rm ex}(n,K_{r},\{M_{s+1},F\})$ behaves very differently from the classical Turán number ${\rm ex}(n,\{M_{s+1},F\})$. The result of (1) in Theorem 1 is also extended to the generalized Turán number ${\rm ex}(n,K_{r},\{M_{s+1},K_{k+1}\})$ by Ma and Hou [18]. Let $\mathcal{N}_{r}(G)$ denote the number of copies of $K_{r}$ in $G$.

Furthermore, they also provided some possible lower bounds for ${\rm ex}(n,K_{r},\{M_{s+1},F\})$ and asked the exact value of ${\rm ex}(n,K_{r},\{M_{s+1},F\})$ when $\chi(F)\geq 3$.

In this paper, we consider the generalized Turán number about the matching and another graph. Before showing our results, we need some definitions. A covering $S$ of $F$ is a subset of $V(F)$ such that $F-S$ is an empty graph, i.e., there is no edge in $F-S$. Let $F$ be a graph and $p$ be an integer, we define a family of subgraphs $\mathcal{F}[p]$ as follow,

In addition to this, we call the covering $S$ an independent covering if $S$ is an independent set in $F$. We also need the definition about the size of the minimum independent covering.

Note that, if $F$ is a bipartite graph, then $p(F)$ is exactly the smallest size of a color class in any proper 2-coloring, as we mentioned in Theorem 2. We determine ${\rm ex}(n,K_{r},\{M_{s+1},F\})$ apart from a constant additive term and some exact values of ${\rm ex}(n,K_{r},\{M_{s+1},F\})$ for special $F$.

Therefore, by Theorem 4, ${\rm ex}(n,\{M_{s+1},F\})=s(n-s)+{\rm ex}(s,\mathcal{F}[s])$ and $G(n,s,\mathcal{F}[s])$ is the unique extremal graph. This corollary extends Theorem 1 and determined the exact value for all non-bipartite graphs and the bipartite graphs with $p(F)\geq s+1$. $\hfill\blacksquare$

Using Theorem 4, we also determine the generalized Turán number ${\rm ex}(n,K_{r},\{M_{s+1},F\})$ when $F$ is color critical.

Since $\chi(F)=k+1$, all graphs in $\mathcal{F}[p]$ have chromatic number at least $k$. If not, the chromatic number of $F$ would not exceed $k$ by the definition of $\mathcal{F}[p]$. Hence we have

$${\rm ex}(p,K_{r-1},\mathcal{F}[p])\geq\mathcal{N}_{r-1}(T_{k-1}(p)),$$

here $T_{k-1}(p)$ denotes the balanced complete ($k-1$)-partite graph on $p$ vertices(called Turán graph).

On the other hand, since $F$ is color critical, we can find a $k+1$-coloring with the color class $V_{1},V_{2},\dots,V_{k+1}$ such that there is only one edge between $V_{1},V_{2}$. Then if we delete the color class $V_{k+1}$, the resulting graph $F^{-}=F-V_{k+1}$ is still color critical and $\chi(F^{-})=k\geq 3$. By the following theorem,

If we take $H=F^{-}$ and $m=r-1$ in the above theorem, then we know when $p\geq c_{0}(F,r)$, ${\rm ex}(p,K_{r-1},F^{-})=\mathcal{N}_{r-1}(T_{k-1}(p))$. Note that $F^{-}\in\mathcal{F}[p]$, then

$$\mathcal{N}_{r-1}(T_{k-1}(p))\leq{\rm ex}(p,K_{r-1},\mathcal{F}[p])\leq{\rm ex}(p,K_{r-1},F^{-})=\mathcal{N}_{r-1}(T_{k-1}(p)).$$

That is to say, ${\rm ex}(p,K_{r-1},\mathcal{F}[p])=\mathcal{N}_{r-1}(T_{k-1}(p))$ is an increasing function when $s\geq p\geq c_{0}(F,r)$. For $p\leq c_{0}(F,r)$, ${\rm ex}(p,K_{r-1},\mathcal{F}[p])$ does not exceed a large constant $C$. Thus we can let $s$ be a large constant depending on $(F,r)$ so that ${\rm ex}(s,K_{r-1},\mathcal{F}[s])$ attains the maximum.

Then by Theorem 4, we know $G(n,s,\mathcal{F}[s])$ is the unique extremal graph. This extends the Theorems 1 and 3. $\hfill\blacksquare$

In this section, we prove Theorem 4. Let $p<\min\{s+1,p(F)\}$ and ${\rm ex}(p,K_{r-1},\mathcal{F}[p])$ attains the maximum at $p=t$. Then $G(n,t,\mathcal{F}[t])$ is $\{M_{s+1},F\}$-free by Definition 1 and

$$\begin{split}{\rm ex}(n,K_{r},\{M_{s+1},F\})&\geq{\rm ex}(t,K_{r-1},\mathcal{F}[t])(n-t)+{\rm ex}(t,K_{r},\mathcal{F}[t])\\ &={\rm ex}(t,K_{r-1},\mathcal{F}[t])n+O(1).\end{split}$$

So the lower bound is done.

Next we prove the upper bound. Let $G$ be the extremal graph of ${\rm ex}(n,K_{r},\{M_{s+1},F\})$. We need the following well-known theorem to discuss the structure of $G$.

Since $G$ is $M_{s+1}$-free, there is a set $B$ satisfying the inequality (1) in the above theorem. Let $G_{1},\dots,G_{m}$ be all components of $G-B$. Note that $s$ is a fixed constant, then most of these components are isolated vertices. Without loss of generality, we may assume the components $G_{1},\dots,G_{\ell}$ are not isolated vertices.

Let $N_{j}$ denote the number of copies of $K_{r}$ which have $j$ vertices in $V(G)-B$ and $r-j$ vertices in $B$. Obviously, $N_{0}\leq\binom{s}{r}$. For other $2\leq j\leq r$, by inequality (1), we have $|B|+\sum_{i=1}^{\ell}|G_{i}|\leq 3s$ and hence

$$\begin{split}N_{j}\leq\sum_{i=1}^{\ell}\mathcal{N}_{j}(G_{i})\binom{s}{r-j}<\binom{3s}{r}=O(1),\end{split}$$

the second inequality holds since we can view $B\cup G_{1}\cup\dots\cup G_{\ell}$ as a big clique. This implies

$\displaystyle\mathcal{N}_{r}(G)=\sum_{j=0}^{r}N_{j}=N_{1}+O(1).$

(2)

Now we mainly deal with the term $N_{1}$. We divide $V(G)\setminus B$ into many subsets by the following way: let $U$ be a subset of $B$,

$$A_{U}=\{v\in V(G)\setminus B:N(v)\cap B=U\}.$$

Let $R=\{U:|A_{U}|\geq|F|\}$ and $Q=\{U:|A_{U}|<|F|\}$. Note that $|R|+|Q|=2^{|B|}$ and we have

$\displaystyle N_{1}=\sum_{U\in R}\mathcal{N}_{r-1}(G[U])|A_{U}|+\sum_{U\in Q}\mathcal{N}_{r-1}(G[U])|A_{U}|.$

(3)

For the set $U$ in $Q$, we have $\mathcal{N}_{r-1}(G[U])|A_{U}|<\binom{|B|}{r-1}|F|$ and hence

$\displaystyle\sum_{U\in Q}\mathcal{N}_{r-1}(G[U])|A_{U}|<2^{|B|}|F|\binom{|B|}{r-1}\leq 2^{s}|F|\binom{s}{r-1}=O(1).$

(4)

For the set $U$ in $R$, since $|A_{U}|\geq|F|$ and $G[U,A_{U}]$ is a complete bipartite graph, we can deduce that $G[U]$ is $\mathcal{F}[|U|]$-free by Definition 1 and $|U|<\min\{s+1,p(F)\}$. Hence

$$\mathcal{N}_{r-1}(G[U])|A_{U}|\leq{\rm ex}(|U|,K_{r-1},\mathcal{F}[|U|])|A_{U}|.$$

On the other hand, as we assumed, $t$ is the integer less than $\min\{s+1,p(F)\}$ such that ${\rm ex}(p,K_{r-1},\mathcal{F}[p])$ attains the maximum, then

$\displaystyle\begin{split}\sum_{U\in R}\mathcal{N}_{r-1}(G[U])|A_{U}|\leq&\sum_{U\in R}{\rm ex}(|U|,K_{r-1},\mathcal{F}[|U|])|A_{U}|\\ \leq&{\rm ex}(t,K_{r-1},\mathcal{F}[t])\sum_{U\in R}|A_{U}|\\ \leq&{\rm ex}(t,K_{r-1},\mathcal{F}[t])(n-|B|).\end{split}$

(5)

Now combine the inequality (2)-(5), we know $\mathcal{N}_{r}(G)={\rm ex}(t,K_{r-1},\mathcal{F}[t])n+O(1)$. We complete the proof of the first part in Theorem 4.

Next we prove the second part, at this time $p(F)\geq s+1$ and ${\rm ex}(p,K_{r-1},\mathcal{F}[p])$ attains the maximum at $p=s$. Furthermore, by the inequality (2)-(5) and the lower bound, we have

$${\rm ex}(s,K_{r-1},\mathcal{F}[s])(n-s)\leq\mathcal{N}_{r}(G)\leq\sum_{U\in R}{\rm ex}(|U|,K_{r-1},\mathcal{F}[|U|])|A_{U}|+O(1).$$

Therefore, when $n$ is large, $B\in R$ with $|B|=s$ and the corresponding $A_{B}$ satisfying $A_{B}=n-O(1)$. Otherwise the right hand of the above would not exceed ${\rm ex}(s,K_{r-1},\mathcal{F}[s])(n-s)$. Hence we know $G[B]$ is $\mathcal{F}[s]$-free since $A_{B}$ is large. This also implies all components of $G-B$ are isolated vertices by Theorem 7. For other vertices in $V(G)-B\cup A_{B}$, we can add all missing edges between them and $B$, this would not create any copy of $F$ since $G[B]$ is $\mathcal{F}[s]$-free, but the number of $K_{r}$ increases. Thus, we know $G=G(n,s,\mathcal{F}(s))$ and we are done. $\hfill\blacksquare$

By the corollaries in Section 2, the only unsolved case for classical Turán number is that $F$ is a bipartite graph with $p(F)\leq s$. In this section we deal with this case. A tree $T[A,B]$ is balanced if $|A|=|B|$. A forest is balanced if each of its component is a balanced tree. The Turán number of balanced forest was studied firstly by Bushaw and Kettle [6] if this forest contains at least two components. Here we study the Turán problem combining the matching and a balanced forest and give a unified proof no matter the forest contains how many components.

Next we prove the upper bound. Let $M=\{v_{1}u_{1},\dots,v_{t}u_{t}\}$ be a maximum matching in $G$, $t\leq s$. This implies $V(G-M)$ is an independent set. We divide $V(G-M)$ into two subsets $W$ and $W^{\prime}$ such that

$$W^{\prime}=\{v\in V(G-M):d(v)\geq p\}~{}\text{and}~{}W=\{v\in V(G-M):d(v)\leq p-1\}.$$

First we claim that $|W^{\prime}|\leq p\binom{2t}{p}$. Indeed, since there are at most $\binom{2t}{p}$ $p$-sets in $V(M)$ and if there are $p\binom{2t}{p}$ vertices in $W^{\prime}$, then by the pigeonhole principle, there are $p$ vertices in $W^{\prime}$ such that they have $p$ common neighbors in $V(M)$. Then we find a large complete bipartite graph, and hence a copy of $F$, a contradiction.

Next we assert that there are at least $2s\binom{2t}{p}$ vertices of degree $p-1$ in $W$. If not, then

$$\begin{split}e(G)\leq&\binom{2t}{2}+2t|W^{\prime}|+2s\binom{2t}{p}(p-1)+(p-2)\left(n-2t-|W^{\prime}|-2s\binom{2t}{p}\right)\\ <&(p-1)(n-p+1).\end{split}$$

The last inequality holds when $n$ is large, a contradiction. This also implies we can find $2s$ vertices of degree $p-1$ in $W$ such that they have $p$ common neighbors in $M$. Without loss of generality, let these $2p$ vertices be $\{x_{1},\dots,x_{2s}\}$ and $U=\{v_{1},\dots,v_{p-1}\}$ be the set of the common neighbors of them.

On the other hand, for all other vertices in $W$ and the vertices in $V(M)-U$ whose degree is at most $p-1$, we can change their neighborhoods to $U$. This operation does not decrease the number of edges and the resulting graph is still $\{F,M_{s+1}\}$-free. Since if there is a copy of $F$ or $M_{s+1}$ in the resulting graph, then we can use the vertices in $\{x_{1},\dots,x_{2s}\}$ to replace the vertices in the copy of $F$ or $M_{s+1}$ which are incident with some new edges. That is, we can find a copy of $F$ or $M_{s+1}$ in $G$, a contradiction.

After the operation, let us redefine the set $W$ and $W^{\prime}$, where $W$ denotes the set of all vertices of degree $p-1$ and have the neighborhood $U$, $W^{\prime}$ denote the other vertices of degree at least $p$. By above, $W^{\prime}$ consists of vertices in the original $W^{\prime}$ and some vertices in $V(M)$ whose degree is at least $p$. Thus $|B|\leq K$ for some constant $K$. Furthermore, $W$ is independent with $|W|\geq n-K-2t=n-O(1)$. Since $G[U,W]$ is a large complete bipartite graph, we have $G[U]$ is $\mathcal{F}[p-1]$-free by the definition.

We also claim that there is no edge between $U$ and $W^{\prime}$. If not, suppose $v_{i}w$ is an edge between $U$ and $W^{\prime}$. Recall that $F=F[A,B]$ is a bipartite graph and $A,B$ are two color classes. There is a vertex, saying $x$, whose all neighbors except one are leaves. Suppose this vertex $x$ is in $B$ and let $y$ be the neighbor of $x$ which is not a leaf. Now we can embed the vertex $x$ into $w$, embed the vertex $y$ into $v_{i}$, embed the other neighbors of $x$ into the neighbor of $w$, embed the other vertices of $A$ into $U$ and the other vertices of $B$ into $W$. This can be done since $|U|=p-1$ and $d(w)\geq p$. Finally, we find a copy of $F$, a contradiction.

Therefore, $W^{\prime}$ induces some connected components of $G$. Furthermore, $G[W^{\prime}]$ is $T_{1}$-free. Otherwise, a copy of $T_{1}$ in $W^{\prime}$ together with a copy of $T_{2}\cup\cdots\cup T_{k}$ in $G[U,W]$ would construct a copy of $F$. Thus, we have

$$\begin{split}e(G)\leq&{\rm ex}(p-1,\mathcal{F}[p-1])+(p-1)(n-p+1-|W^{\prime}|)+{\rm ex}(|W^{\prime}|,T_{1})\\ \leq&{\rm ex}(p-1,\mathcal{F}[p-1])+(p-1)(n-p+1)-|W^{\prime}|(p-1)+\frac{v(T_{1})-2}{2}|W^{\prime}|\\ \leq&{\rm ex}(p-1,\mathcal{F}[p-1])+(p-1)(n-p+1).\end{split}$$

The last inequality holds under the assumption of Erdős-Sós conjecture. From the above inequalities, if $F$ is a real forest, then $(p-1)>(v(T_{1})-2)/2$. So if the equality holds, then $W^{\prime}=\emptyset$ and $G=G(n,p-1,\mathcal{F}[p-1])$. If $F$ is a tree, then the equality holds if and only if $W^{\prime}$ induces some disjoint cliques $K_{2p-1}$. But since $G$ is $M_{s+1}$-free, $W^{\prime}$ induces at most $\frac{s-p+1}{p-1}$ copies of $K_{2p-1}$. That is $G=G(n-(p-1)-t(2p-1),p-1,\mathcal{F}[p-1])\cup tK_{2p-1}$ with $t\leq\frac{s-p+1}{p-1}$. The proof is completed.$\hfill\blacksquare$

This Research was supported by NSFC under grant numbers 12161141003 and 11931006.