Advances on the Conjecture of Erd\H{o}s-S\'os for spiders

TL;DR

This paper investigates conditions under which certain graphs contain all possible spiders of a given size, advancing the Erdős–Sós conjecture for specific classes of graphs and spider configurations.

Contribution

It provides new sufficient conditions for graphs to contain all spiders of size up to 10, including results for Hamiltonian and 2-connected graphs, extending the Erdős–Sós conjecture.

Findings

Hamiltonian graphs with e(G)>n(k-1)/2 contain any k-spider.

Graphs with average degree > k-1 contain all spiders of size k for k ≤ 10.

2-connected graphs with average degree > sum of legs contain specific 4-leg spiders.

Abstract

- A hamiltonian graph $G$ verifying $e(G)>n(k-1)/2$ %with a vertex of degree greater or equal than $k$ contains any $k$-spider. - If $G$ is a graph with average degree $\bar{d} > k-1$, then every spider of size $k$ is contained in $G$ for $k\le 10$. - A $2$-connected graph with average degree $\bar{d} > \ell_2+\ell_3+\ell_4$ contains every spider of $4$ legs $S_{1,\ell_2,\ell_3,\ell_4}$. We claim also that the condition of $2$-connection is not needed, but the proof is very long and it is not included in this document.

Full text

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A hamiltonian graph $G$ verifying $e(G)>n(k-1)/2$ contains any $k$-spider.

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If $G$ is a graph with average degree $\bar{d}>k-1$, then every spider of size $k$ is contained in $G$ for $k\leq 10$.

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A $2$-connected graph with average degree $\bar{d}>\ell_{2}+\ell_{3}+\ell_{4}$ contains every spider of $4$ legs $S_{1,\ell_{2},\ell_{3},\ell_{4}}$. We claim also that the condition of $2$-connection is not needed, but the proof is very long and it is not included in this document.

1 Introduction

The Erdős-Sós conjecture [2] says that a graph $G$ on $n$ vertices and number of edges $e(G)>n(k-1)/2$ contains all trees of size $k$.

By $g(n,k)$ Erdős and Gallai [3] denoted the maximum number of edges of a graph $G$ on $n$ vertices containing no cycles with more than $k$ edges. Moreover, these authors proved that $g(n,k)\leq\frac{1}{2}(n-1)k,\mbox{ for }2\leq k\leq n.$ (Theorem 2.7) Thus if $e(G)>(n-1)k/2$ then $G$ contains a cycle with at least $k+1$ edges.

Fan and Sun [4] used Theorem 2.7 to note that every graph $G$ with $e(G)>n(k-1)/2$ has a circumference of length at least $k$. This is clear because $e(G)>n(k-1)/2>(n-1)(k-1)/2$. Then they used this observation to prove that every graph with $e(G)>n(k-1)/2$ contains any $k$-spider of three legs. We will prove that a hamiltonian graph $G$ with $e(G)>n(k-1)/2$ contains any $k$-spider. We will prove also that a connected graph with $e(G)>n(k-1)/2$ contains all spider of four legs, with one leg of unity length.

2 Results

We need to introduce the following notation. Let $S_{\ell_{1},\ldots,\ell_{f}}$ be a $k$-spider of $f$ legs of lengths $\ell_{1},\dots,\ell_{f}$, i.e., $\ell_{1}+\dots+\ell_{f}=k$. Let $P_{1},\dots,P_{f}$ be the $f$ legs of the $k$-spider such that $e(P_{i})=\ell_{i}$, $i=1,2,\dots,f$. We may assume that $\ell_{1}\leq\ell_{2}\leq\cdots\leq\ell_{f}$ and if the spider has $4$ legs ($f=4$) and $\ell_{1}=1$ then $\ell_{2}\geq 2$ because if $\ell_{2}=1$, then $S_{1,1,\ell_{3},\ell_{4}}$ is a caterpillar, which is included in any graph $G$ with $e(G)>|V(G)|(k-1)/2$ where $k=\ell_{3}+\ell_{4}+2$ [1]. Note that $\ell_{1}\leq k/f$ and $\ell_{2}\leq(k-\ell_{1})/(f-1)$. (If $f=4$, $\ell_{1}\leq k/4$ and $\ell_{2}\leq(k-\ell_{1})/3$).

In general, $\ell_{i}\leq k-\sum\limits_{j=1}^{i-1}\ell_{j}/(f-i+1)=\sum\limits_{j=i}^{f}\ell_{j}/(f-i+1)$

Let $G$ be a graph with $e(G)>|V(G)|(k-1)/2$. Let $H$ be a minimal induced subgraph of $G$ such that $e(H)>|V(H)|(k-1)/2$. By the minimality, $H$ is connected and $deg_{H}(v)\geq k/2$ for every $v\in V(H)$. If $H$ has a copy of $S_{\ell_{1},\ell_{2},\dots,\ell_{f}}$, so does $G$.

Theorem 2.1

Let $G$ be a graph and $H$ a hamiltonian subgraph of $G$. Suppose that there exists a vertex $x_{0}\in V(H)$ such that $deg_{H}(x_{0})\geq k$. Then $H$ contains (and so $G$) any $k$-spider.

Proof  Let $m=|V(H)|$ and $x_{0}x_{1}\cdots x_{m-1}x_{0}$ a hamiltonian cycle of $H$. We will prove the theorem by induction on $k$.

For $k=2$, it is easy to check that theorem holds, because the only $2$-spider are $S_{1,1}$ and $S_{2}$ and they are isomorphic to the path of length $2$, which is contained in the hamiltonian cycle. Moreover $S_{1,1}$ or $S_{2}$ can be taken with root $x_{0}$. For $k=3$, the only $3$-spiders are $S_{1,1,1}$ that is contained in $H$ with root $x_{0}$ because $deg_{H}(x_{0})\geq k=3$, and the isomorphic spiders $S_{1,2}$ and $S_{3}$ which clearly are contained in $H$ with $x_{0}$ as root.

Suppose that theorem is true for every $k^{\prime}$ with $3<k^{\prime}<k$, and let us show that the theorem is also valid for $k$. Let $S_{\ell_{1},\ell_{2},\dots,\ell_{f}}$ be a spider of $f$ legs and size $k$, i.e, $\ell_{1}+\ell_{2}+\dots+\ell_{f}=k$. Let $\alpha$ be the smallest index such that $x_{\alpha}\in N(x_{0})$ with $\alpha\geq\ell_{1}+1$, that there exists because $deg_{H}(x_{0})\geq k>\ell_{1}$. Let $H^{\prime}\subset H$ be the subgraph induced by $V(H)\setminus\{x_{1},\dots,x_{\alpha-1}\}$. Clearly $H^{\prime}$ is hamiltonian because $C=x_{0}x_{\alpha}x_{\alpha+1}\cdots x_{n-1}x_{0}$ is a hamiltonian cycle of $H^{\prime}$. Moreover $deg_{H^{\prime}}(x_{0})\geq k-\ell_{1}$. By the inductive hypothesis on $k$, $H^{\prime}$ contains all the spiders with root in $x_{0}$ and size $k-\ell_{1}$. Particularly, $H^{\prime}$ contains the spider $S_{\ell_{2},\dots,\ell_{f}}$ whose legs are denoted by $P_{2},\dots,P_{f}$. Then, the spider with root $x_{0}$ and legs $P_{1}=x_{0},x_{1},\dots,x_{\ell_{1}}$, $P_{2},\dots,P_{f}$ is contained in $H$. Thus $S_{\ell_{1},\ell_{2},\dots,\ell_{f}}$ is contained in $G$, finishing the proof.

Observe that $e(G)>|V(G)|(k-1)/2$ is equivalent to the requirement that the average degree $\overline{d}>k-1$ and so maximum degree of $G$, $\Delta(G)\geq\overline{d}>k-1$.

Corollary 2.1

Let $G$ be a hamiltonian graph with average degree $\bar{d}>k-1$. Then every spider of size $k$ is contained in $G$.

Lemma 2.1

Let $G$ be a 2-connected graph with average degree $\bar{d}>k-1$. Then every vertex of degree at least $k$ lies on a cycle $C_{s}$ of length $s\geq k$.

Proof  Since $\bar{d}>k-1$ then $\Delta(G)\geq k$. The results follows directly from Theorem 1.16 in [3].

Corollary 2.2

Let $G$ be a graph with average degree $\bar{d}>k-1$. Then every spider of size $k$ is contained in $G$ for $k\leq 9$.

Proof  Every spider with legs of length at most 4 are contained in $G$ by Theorem 4.1 of [4]. Moreover, every spider with three legs are contained in $G$ by Theorem 3.1 of [4]. Therefore the remaining spiders are the comet $S_{1,1,1,1,5}$ and the caterpillar $S_{1,1,2,5}$ and therefore the result is valid by [1].

Theorem 2.2

Let $G$ be a 2-connected graph with average degree $\bar{d}>k-1$. Then $G$ contains every $k$-spider $S_{1,\ell_{2},\ell_{3},\ell_{4}}\ (k=1+\ell_{2}+\ell_{3}+\ell_{4})$.

Proof  We will suppose that $2\leq\ell_{2}\leq\ell_{3}\leq\ell_{3}$ (otherwise $S_{1,1,\ell_{3},\ell_{4}}$ is a caterpillar and is contained in $G$). Let $x_{0}\in V(G)$ be with $deg_{G}(x_{0})=\Delta(G)\geq k$. By Lemma 2.1, we can take a cycle $C_{s}$ of maximum length $|C_{s}|=s\geq k$ such that $x_{0}\in V(C_{s})$. Let $C_{s}=x_{0}x_{1}\cdots x_{s-1}x_{0}$. If $N(x_{0})\subset V(C_{s})$, the subgraph $H$ of $G$ induced by the vertices of $C_{s}$ is clearly hamiltonian and has a vertex $x_{0}$ of Therefore by Theorem 2.1, $H$ (and so $G$) contains all spiders of size $k$ and particularly $S_{1,\ell_{2},\ell_{3},\ell_{4}}$.

Hence assume that $N(x_{0})\not\subset V(C_{s})$. Therefore we can consider a path $P=x_{0}u_{1}\cdots u_{\ell}$ starting in $x_{0}$ of maximum length such that $V(C_{s})\cap V(P)=\{x_{0}\}$. Two cases need to be distinguished according to $\ell\geq\ell_{2}$ or $\ell<\ell_{2}$.

Case 1: $\ell\geq\ell_{2}$.

If there exists $y\notin V(C_{s})\cup V(P)$ such that $y\in N(x_{0})$, then $S_{1,\ell_{2},\ell_{3},\ell_{4}}$ is contained in $G$ and its legs are $P_{1}=x_{0}y$, $P_{2}=x_{0}u_{1}\cdots u_{\ell_{2}}$, $P_{3}=x_{0}x_{1}\cdots x_{\ell_{3}}$ and $P_{4}=x_{0}x_{s-1}\cdots x_{s-\ell_{4}}$ (see Figure 3). Therefore, assume $N(x_{0})\subset V(C_{s})\cup V(P)$ and let us study the following subcases.

1. (a)

If $u_{m}\in N(x_{0})$ with $2\leq m\leq\ell-\ell_{2}+1$ , then $G$ contains the spider $S_{1,\ell_{2},\ell_{3},\ell_{4}}$, of legs $P_{1}=x_{0}u_{1}$, $P_{2}=x_{0}u_{m}u_{m+1}\cdots u_{m+\ell_{2}-1}$, $P_{3}=x_{0}x_{1}\cdots x_{\ell_{3}}$ and $P_{4}=x_{0}x_{s-1}\cdots x_{s-\ell_{4}}$ (see Figure 3). 2. (b)

If $u_{m}\in N(x_{0})$ with $\ell_{2}+1\leq m$, then $G$ contains the spider $S_{1,\ell_{2},\ell_{3},\ell_{4}}$ of legs $P_{1}=x_{0}u_{1}$, $P_{2}=x_{0}u_{m}u_{m-1}\cdots u_{m-\ell_{2}+1}$, $P_{3}=x_{0}x_{1}\cdots x_{\ell_{3}}$ and $P_{4}=x_{0}x_{s-1}\cdots x_{s-\ell_{4}}$ (see Figure 3). 3. (c)

Otherwise we must distinguish between two different situations. If $\ell-\ell_{2}+2\leq\ell_{2}$ then $(N(x_{0})-u_{1})\cap V(P)\subseteq\{u_{\ell-\ell_{2}+2},\dots,u_{\ell_{2}}\}$, and $|N(x_{0})\cap V(P)|\leq\ell_{2}-(\ell-\ell_{2}+2)+1=2\ell_{2}-\ell-1\leq 2\ell_{2}-\ell_{2}-1=\ell_{2}-1$. By the contrary, if $\ell-\ell_{2}+2>\ell_{2}$ then $N(x_{0})\cap V(P)=\{u_{1}\}$ and $|N(x_{0})\cap V(P)|=1\leq\ell_{2}-1$. Therefore, in both cases, $|N(x_{0})\cap V(C_{s})|\geq k-1-\ell_{2}+1=k-\ell_{2}=1+\ell_{3}+\ell_{4}$. Thus there must exist an edge $x_{0}x_{h}$ with $\ell_{3}<h<s-\ell_{4}$ so that $G$ contains the spider $S_{1,\ell_{2},\ell_{3},\ell_{4}}$ of legs $P_{1}=x_{0}x_{h}$, $P_{2}=x_{0}u_{1}\cdots u_{\ell_{2}}$, $P_{3}=x_{0}x_{1}\cdots x_{\ell_{3}}$ and $P_{4}=x_{0}x_{s-1}\cdots x_{s-\ell_{4}}$ (see Figure 3).

Case 2: $\ell<\ell_{2}$.

Note that $N(u_{\ell})\subset V(C_{s})\cup V(P)$ as $P$ has maximum length. Since $|N(u_{\ell})\cap V(P)|\leq\ell\leq\ell_{2}-1$, $N(u_{\ell})\not\subset V(P)$ because $\ell_{2}\leq(k-1)/3<k/2\leq deg(u_{\ell})$. Note also that since $C_{s}$ has maximum length, $N(u_{\ell})\cap\{x_{1},\ldots,x_{\ell}\}=\emptyset$, otherwise the cycle $x_{0}u_{1}\cdots u_{\ell}x_{i}x_{i+1}\cdots x_{s-1}x_{0}$ would have a length greater than $s$ which is a contradiction. Similarly, $N(u_{\ell})\cap\{x_{s-\ell},\ldots,x_{s-1}\}$ $=\emptyset$, otherwise the cycle $x_{0}u_{1}\cdots u_{\ell}x_{i}x_{i-1}\cdots x_{1}x_{0}$ would have a length greater than $s$ again a contradiction (see Figure 6).

We may notice that if $x_{j},x_{j+1}\in V(C_{s})$ and $x_{j}\in N(u_{\ell})$, then $x_{j+1}\not\in N(u_{\ell})$ because $C_{s}$ has maximum length. Since $|\{x_{\ell+1},\dots,x_{\ell_{2}}\}|=|\{x_{s-\ell_{2}},\dots,x_{s-\ell-1}\}|=\ell_{2}-\ell$ then $|N(u_{\ell})\cap(\{x_{\ell+1},\dots,x_{\ell_{2}}\}\cup\{x_{s-\ell_{2}},\dots,x_{s-\ell-1}\})|\leq 2\left\lceil(\ell_{2}-\ell)/2\right\rceil$. Since $|N(u_{\ell})\cap V(P)|\leq\ell$, it follows that $|N(u_{\ell})\cap\{x_{\ell_{2}+1},\dots,x_{s-\ell_{2}-1}\}|\geq k/2-2\left\lceil(\ell_{2}-\ell)/2\right\rceil-\ell\geq k/2-\ell_{2}-1>0$ (because $\ell\leq\ell_{2}-1$). Let $\alpha$ be the smallest index such that $x_{\alpha}\in N(u_{\ell})\cap\{x_{\ell_{2}+1},\dots,x_{s-\ell_{2}-1}\}$. Since $k/2-\ell_{2}-1\leq\left|N(u_{\ell})\cap\{x_{\ell_{2}+1},\dots,x_{s-\ell_{2}-1}\}\right|=\left|N(u_{\ell})\cap\{x_{\alpha},\dots,x_{s-\ell_{2}-1}\}\right|\leq\left\lceil(s-\ell_{2}-\alpha)/2\right\rceil\leq(s-\ell_{2}-\alpha+1)/2$ then $\alpha\leq s-k+\ell_{2}+3$. But the inequality does not hold because it would mean that $2\left\lceil(\ell_{2}-\ell)/2\right\rceil=\ell_{2}-\ell+1$, so $u_{\ell}$ would be adjacent to $x_{s-\ell_{2}}$, and $\left\lceil(s-\ell_{2}-\alpha)/2\right\rceil=(s-\ell_{2}-\alpha+1)/2$ and $u_{\ell}$ would be adjacent to $x_{s-\ell_{2}-1}$ too, and that it is not possible because of the maximality of the cycle. So $\alpha<s-k+\ell_{2}+3$.

If there exists $y\notin V(C_{s})\cup V(P)$ such that $y\in N(x_{0})$, then the spider $S_{1,\ell_{2},\ell_{3},\ell_{4}}$ with legs $P_{1}=x_{0}y$, $P_{2}=x_{0}x_{1}\cdots x_{\ell_{2}}$, $P_{3}=x_{0}u_{1}\cdots u_{\ell}x_{\alpha}\cdots x_{\alpha+\ell_{3}-\ell-1}$ and $P_{4}=x_{0}x_{s-1}\cdots x_{s-\ell_{4}}$ is contained in $G$ because $\alpha+\ell_{3}-\ell-1<s-k+\ell_{2}+3+\ell_{3}-\ell-1=s-\ell_{4}+1-\ell\leq s-\ell_{4}$ (see Figure 6). If $N(x_{0})\subset V(C_{s})\cup V(P)$, since $|N(x_{0})\cap V(P)|\leq\ell$, it follows that $|N(x_{0})\cap V(C_{s})|\geq k-\ell$. As the index set $I=\{1,\ldots,\ell_{2},\alpha,\ldots,{\alpha+\ell_{3}-\ell-1},{s-\ell_{4}},\ldots,{s-1}\}$ has cardinality $k-\ell-1$, there must exist $x_{q}\in N(x_{0})$ ($q\not\in I$). As $\alpha+\ell_{3}-\ell-1<s-\ell_{4}$, the spider of legs $P_{1}=x_{0}x_{q}$, $P_{2}=x_{0}x_{1}\cdots x_{\ell_{2}}$, $P_{3}=x_{0}u_{1}\cdots u_{\ell}x_{\alpha}\cdots x_{\alpha+\ell_{3}-\ell-1}$ and $P_{4}=x_{0}x_{s-1}\cdots x_{s-\ell_{4}}$ is contained in $G$ (see Figure 6).

Theorem 2.3

If $G$ is a connected graph with average degree $\bar{d}>k-1$, then $G$ contains every $k$-spider $S_{1,\ell_{2},\ell_{3},\ell_{4}}\ (k=1+\ell_{2}+\ell_{3}+\ell_{4})$.

Proof is similar to the proof of Theorem 2.2, but is very long and it is not included here.

Corollary 2.3

Let $G$ be a graph with average degree $\bar{d}>k-1$. Then every spider of size $k$ is contained in $G$ for $k\leq 10$.

Proof  Every spider with legs of length at most 4 are contained in $G$ by Theorem 4.1 of [4]. Moreover, every spider with three legs are contained in $G$ by Theorem 3.1 of [4]. For the caterpillars $S_{1,1,1,1,1,5}$, $S_{1,1,1,1,6}$, $S_{1,1,1,2,5}$, $S_{1,1,1,7}$, $S_{1,1,2,6}$ or $S_{1,1,3,5}$ the result is valid by [1]. It remains the spider $S_{1,2,2,5}$, which can be applied the Theorem 2.3.