The Turán number of star forests

Abstract

The Turán number of a graph H, denoted by ex(n, H), is the maximum number of edges in any graph on n vertices containing no H as a subgraph. Let S_ℓ denote the star on $\ell +1$ vertices and let k · S_ℓ denote disjoint union of k copies of S_ℓ. In this paper, for appropriately large n, we determine the Turán numbers for k · S_ℓ and F, where F is a forest with components each of order 4, which improve the results of Lidický et al. (2013).

Introduction

All graphs are simple and our notation is standard (see, e.g., [1]). Let K_n, P_n, $S_{n-1}$ denote the complete graph, path and star on n vertices, respectively. Given a graph G, we use e(G) to denote the number of edges. For a vertex v ∈ V(G), let N_G(v) denote the set of vertices in G which are adjacent to v. The degree of a vertex v ∈ V(G), denoted d_G(v), is the number of vertices in N_G(v). Moreover, we define $N_G\left[v\right]=N_G\left(v\right)\cup \left\{v\right\}$ and let Δ(G) denote the maximum degree. Without creating confusion, we simple write d(v), N(v), N[v], respectively. Given two vertex-disjoint graphs G and H, let G ∪ H denote the disjoint union of graphs G and H, k · G the disjoint union of k copies of G, and $G+H$ the graph obtained from G ∪ H by joining all vertices of G to all vertices of H. We use $\overline{G}$ to denote the complement of G. For any set S ⊂ V(G), let G[S] (resp. G\S) denote the subgraph of G induced by S (resp. V(G)\S) and let |S| denote the cardinality of S. For a graph G and its subgraph H, let $G-H$ denote the subgraph induced by V(G)\V(H). For any positive integer k, let $\left[k\right]:=\{1,2,\dots ,k\}$.

A graph is F-free if it does not contain a copy of F as a subgraph. The Turán number of a graph F, denoted ex(n, F), is the maximum number of edges in an F-free graph on n vertices. As one of the well-known results in extremal graph theory, Turán theorem [11], [12] gives that the maximum number of edges in a K_r-free graph on n vertices is $(1-1/(r-1))n^2/2$ and the corresponding extremal graph is the balanced complete r-partite graph on n vertices which is also called Turán graph $T_{r-1}\left(n\right)$. This extends the Mantel theorem, which proved that $ex(n,K_3)=\lfloor n^2/4\rfloor$. When K_r is replaced by any arbitrary graph H with at least one edge, the celebrated Erdös-Stone theorem states that the maximum number of edges is $(1-1/(\chi \left(H\right)-1))n^2/2+o\left(n^2\right),$ where χ(H) denotes the chromatic number of H.

A natural generalization of this problem is to determine the Turán number of disjoint union of some graphs. Let $M_k=k·K_2$ denote the matching of size k. Erdös and Gallai [5] in 1959 studied the Turán number of matchings and paths, showing that $ex(n,M_k)=\max \{\left(\genfrac{}{}{0pt}{}{2k-1}{2}\right),\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+(k-1)(n-k+1)\}$ and $ex(n,P_k)\le (k-2)n/2$. Later, Moon [9] and Simonovits [10] extended this to the case k · K_r, showing that for sufficiently large n, $K_{k-1}+T_{r-1}(n-k+1)$ is the unique extremal graph on n vertices. On the other hand, Erdös and Sós [4], [6] also conjectured that every graph on n vertices with $ex(n,M_k)+1$ edges contains all forests with k edges. This was partially answered by Brandt [2], who proved that any graph G with $e\left(G\right)\ge ex(n,M_k)+1$ contains every forest with k edges containing no isolated vertices.

Recently, Gorgol [7] further determined the lower and upper bound for Turán numbers for disjoint union of arbitrary connected graphs. We state the lower bound for ex(m, p · G) below.

Theorem 1.1 [7]

Let n, p, m be positive integers with m ≥ pn and G be a connected graph on n vertices. Then $ex(m,p·G)\ge \max \{ex(m-pn+1,G)+\left(\genfrac{}{}{0pt}{}{pn-1}{2}\right),ex(m-p+1,G)+(p-1)m-\left(\genfrac{}{}{0pt}{}{p}{2}\right)\}$.

A linear forest (resp. star forest) is a forest whose connected components are paths (resp. stars). Bushaw and Kettle [3] determined the Turán number of k · P_ℓ for sufficiently large n. This result was extended by Lidický et al. [8], who determined the Turán number of arbitrary linear forests for sufficiently large n. Moreover, Yuan and Zhang [14] completely determined the Turán number of the linear forests containing at most one odd path for all n. It seems non-trivial to determine ex(n, F_m) for all n, where F_m is a linear forest containing at least two odd paths. Recently, Yuan and Zhang [14], [15] determined the value of ex(n, H), where $H\in \{k·P_3,P_3\cup P_{2\ell +1},2·P_5\}$ for all n. We state the above theorem here since it is an important tool for our proof. Let EX(n, G) denote a G-free graph on n vertices with ex(n, G) edges.

Theorem 1.2 [14]

Let n, k be any integers and ℓ even integer. Then

$$ex(n,k·P_{\ell })=\max \{\left(\genfrac{}{}{0pt}{}{k\ell -1}{2}\right)+\frac{(\ell -2)(n-k\ell +1)}{2},\left(\genfrac{}{}{0pt}{}{\frac{k\ell }{2}-1}{2}\right)+(\frac{k\ell }{2}-1)(n-\frac{k\ell }{2}+1)\}.$$

The extremal graphs are $EX(n-k\ell +1,P_{\ell })\cup K_{k\ell -1}$ and $K_{k\ell /2-1}+\overline{K_{n-k\ell /2+1}}.$

To determine the Turán number of star forests, Lidický et al. [8] proved the following result.

Theorem 1.3 [8]

Let $F_k=S_{{\ell }_1}\cup S_{{\ell }_2}\cup \dots \cup S_{{\ell }_k}$ and ℓ₁ ≥ ℓ₂ ≥ ⋅⋅⋅ ≥ ℓ_k be integers. For sufficiently large n,

$$ex(n,F_k)=\underset{1\le i\le k}{\max }\{(i-1)(n-i+1)+\left(\genfrac{}{}{0pt}{}{i-1}{2}\right)+\lfloor \frac{{\ell }_i-1}{2}(n-i+1)\rfloor \}.$$

Moreover, the extremal graph is $K_{i-1}+EX(n-i+1,S_{{\ell }_i})$ for some i ∈ [k].

Yin and Rao [13] improved the result of Lidický et al. by determining the above result when ${\ell }_1=\cdots ={\ell }_k=\ell$ is true for $n\ge {\ell }^{2}k(k-1)/2+(k-2)+\max \{k\ell ,{\ell }^2+2\ell \}$.

Furthermore, motivated by these results, we prove the following theorem.

Theorem 1.4

For k ≥ 2, ℓ ≥ 3, and $n\ge k({\ell }^2+\ell +1)-\frac{\ell }{2}(\ell -3),$

$$ex(n,k·S_{\ell })=\left(\genfrac{}{}{0pt}{}{k-1}{2}\right)+(k-1)(n-k+1)+\lfloor \frac{\ell -1}{2}(n-k+1)\rfloor .$$

Moreover, the extremal graph is $K_{k-1}+EX(n-k+1,S_{\ell })$.

In particular, for the case of $k=2$ and $\ell =3,$ we determine the exact value of ex(n, 2 · S₃) for all n.

Theorem 1.5

For all n, $ex(n,2·S_3)=\max \{n+14,2(n-1)\}.$

Moreover, Lidický et al. [8] investigated the Turán number of forests F, where $F=k_1·P_4\cup k_2·S_3$ with k₁ ≥ 1. They proved that for sufficiently large n ($n=k_2+3d+r$), $ex(n,F)=\max \{e\left(G_1\right),e\left(G_2\right)\},$ where $G_1=K_{k_2}+(d·K_3\cup K_r)$ and $G_2=K_{2k_1+k_2-1}+\overline{K_{n-2k_1-k_2+1}}$. We continue to study this case and prove that the extremal graph is the same when $n\ge 10k_1+13k_2+3$.

Theorem 1.6

Let $F=k_1·P_4\cup k_2·S_3$ (k₁ ≥ 1), and $n=k_2+3d+r\ge 10k_1+13k_2+3$ with r ≤ 2. Then $ex(n,F)=\max \{e\left(G_1\right),e\left(G_2\right)\},$ where $G_1=K_{k_2}+(d·K_3\cup K_r)$ and $G_2=K_{2k_1+k_2-1}+\overline{K_{n-2k_1-k_2+1}}$. Furthermore, the extremal graph is G₁ when $k_1=1$ and G₂ when k₁ > 1. In particular, G₂ is also a extremal graph when $k_1=1$ and r ≠ 0.