On the Ramsey-Goodness of Paths

Abstract

For a graph G, we denote by $\nu (G)$ the order of G, by $\chi (G)$ the chromatic number of G and by $\sigma (G)$ the minimum size of a color class over all proper $\chi (G)$-colorings of G. For two graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is the least integer r such that for every graph G on r vertices, either G contains a $G_1$ or $\overline{G}$ contains a $G_2$. Suppose that $G_1$ is connected. We say that $G_1$ is $G_2$-good if $R(G_1,G_2)=(\chi (G_2)-1)(\nu (G_1)-1)+\sigma (G_2)$. In this note, we obtain a condition for graphs H such that a path is H-good.

1 Introduction

Throughout this paper, all graphs are finite and simple. Let $G_1$ and $G_2$ be two graphs. The Ramsey number $R(G_1,G_2)$, is defined as the least integer r such that for every graph G on r vertices, either G contains a $G_1$ or $\overline{G}$ contains a $G_2$, where $\overline{G}$ is the complement of G.

We denote by $\nu (G)$ the order of G, by $\delta (G)$ the minimum degree of G, by $\omega (G)$ the component number of G, by $\chi (G)$ the chromatic number of G and by $\sigma (G)$ the minimum size of a color class over all proper $\chi (G)$-colorings of G. For two disjoint graphs $G_1$ and $G_2$, the union of $G_1$ and $G_2$ is defined as $V(G_1\cup G_2)=V(G_1)\cup V(G_2)$ and $E(G_1\cup G_2)=E(G_1)\cup E(G_2)$; and the join of $G_1$ and $G_2$ is defined as $V(G_1\vee G_2)=V(G_1)\cup V(G_2)$, and $E(G_1\vee G_2)=E(G_1+G_2)\cup \{xy:x\in V(G_1),y\in V(G_2)\}$. The union of k disjoint copies of the same graph G is denoted by kG.

Theorem 1

(Burr [5]) For all graphs G and H, with G connected and $\nu (G)\ge \sigma (H)$,

$$\begin{aligned} R(G,H)\ge (\chi (H)-1)(\nu (G)-1)+\sigma (H). \end{aligned}$$

We say G is H-good if $R(G,H)=(\chi (H)-1)(\nu (G)-1)+\sigma (H)$. Lin et al. proved the following theorem on Ramsey-goodness of trees.

Theorem 2

(Lin et al. [11]) Let T be a tree and H be a graph. If T is H-good and $\sigma (H)=1$, then T is also $K_1\vee H$-good.

By $P_n$ and $C_n$ we denote the path and cycle on n vertices, respectively. For the case of T being a path, we get an extension of Theorem 2.

Theorem 3

Let $n\ge 2$, and H be a subgraph of $K_{a_1,a_2,\ldots ,a_k}$, $k=\chi (H)$, such that

$$\begin{aligned} a_i\le \left\lceil \frac{k(n-1)+1}{k+i}\right\rceil , 1\le i\le k, \end{aligned}$$

If $P_n$ is H-good, then $P_n$ is also $K_1\vee H$-good.

Note that H is a subgraph of $K_{a_1,a_2,\ldots ,a_k}$ if and only if there is a proper coloring of H such that the size of the $i\hbox {th}$ color class is at most $a_i$, $1\le i\le k$.

We prove Theorem 3 in Sect. 3. In Sect. 2, we will apply Theorem 3 to show some results of Ramsey values involving paths.

Note that $\sigma (K_1\vee H)=1$. By Theorem 2, we can see that under the conditions of Theorems 2 and 3, $P_n$ is $K_t\vee H$-good for all $t\ge 1$.

Theorem 4

Let $n\ge 3$ and H be a graph with $\sigma (H)\le 2$. If $P_n$ is H-good, then $P_n$ is also $(2K_1\vee H)$-good.

Proof

Since $P_n$ is H-good, $R(P_n,H)=(\chi (H)-1)(n-1)+\sigma (H)$. Set

$$\begin{aligned} r=(\chi (2K_1\vee H)-1)(n-1)+\sigma (2K_1\vee H)=\chi (H)(n-1)+\sigma (H). \end{aligned}$$

Then $r=R(P_n,H)+n-1$.

From Theorem 1, we have $R(P_{n},2K_1\vee H)\ge r$. Now let G be an arbitrary graph of order r without $P_{n}$ as a subgraph. We will prove that $\overline{G}$ contains $2K_1\vee H$. Let $P=v_1v_2\ldots v_k$ be a longest path of G. Thus $k\le n-1$.

If $k=1$ then G is an empty graph and $\overline{G}$ is complete. Note that $\nu (G)=r\ge R(P_{n},H)+2\ge \nu (H)+2$. So $\overline{G}$ contains $2K_1\vee H$.

Now we assume that $2\le k\le n-1$. Let $G'$ be a subgraph of G induced by $V(G)-V(P)$. Then $\nu (G')=r-k\ge R(P_{n},H)$. So $\overline{G'}$ contains H. Note that $v_1$ and $v_k$ are nonadjacent to every vertex of $G'$. Thus $\overline{G}$ contains $2K_1\vee H$. $\square$

From Theorem 4 we get the following result.

Corollary 1

Let $n\ge 3$ and H be a graph with $\sigma (H)\le 2$. If $P_n$ is H-good, then $P_n$ is also $(\overline{tK_2}\vee H)$-good.

2 Some Corollaries

In this section, we will list some known results for the Ramsey numbers involving paths. After each result, we apply Theorems 2, 3 and 4 to get a new Ramsey numbers involving paths. We denote by $L_s^t$ ($s\ge t+1$) the graph obtained from $K_{s+t}$ by removing the edges of a matching of size t, i.e., $L_s^t=\overline{tK_2}\vee K_{s-t}$. We use $\mathrm{par}(m)$ to denote the parity of m. In the following corollaries, we always assume that $n\ge 3$.

Theorem 5

(Gerencsér and Gyárfás [10]) If $2\le m\le n$, then

$$\begin{aligned} R(P_n,P_m)=n+\left\lfloor \frac{m}{2}\right\rfloor -1. \end{aligned}$$

Corollary 2

Let $t\ge 0$ be an integer. If $2\le m\le n$, then

$$\begin{aligned} R(P_n,L_s^t\vee P_m)=(s+1)(n-1)+1. \end{aligned}$$

Proof

By Theorem 5, $P_n$ is $P_m$-good. Take $a_1=\lceil m/2\rceil$ and $a_2=\lfloor m/2\rfloor$. By Theorem 3, $P_n$ is $(K_1\vee P_m)$-good. So by Theorems 2 and 4 we get the assertion. $\square$

The kipas $\widehat{K}_m$ is the graph obtained by joining a $K_1$ and a path $P_m$. For the case $(s,t)=(1,0)$ in Corollary 2, we can get the values of path-kipas Ramsey numbers $R(P_n,\widehat{K}_m)$ for $3\le m\le n$, which was already obtained by Saleman and Broersma [16].

Theorem 6

(Faudree et al. [9]) If $n\ge 2$ and $m\ge 3$, then

$$\begin{aligned} R(P_n,C_m)=\left\{ \begin{array}{ll} 2n-1, &{} n\ge m \text{ and } m \text{ is } \text{ odd };\\ n+m/2-1, &{} n\ge m \text{ and } m \text{ is } \text{ even };\\ \max \{m+\lfloor n/2\rfloor -1,2n-1\}, &{} m>n \text{ and } m \text{ is } \text{ odd };\\ m+\lfloor n/2\rfloor -1, &{} m>n \text{ and } m \text{ is } \text{ even }. \end{array}\right. \end{aligned}$$

Corollary 3

Let $t\ge 0$ be an integer. If m is even, $4\le m\le n$; or m is odd and $3\le m\le \lceil 3n/2\rceil$, then

$$\begin{aligned} R(P_n,L_s^t\vee C_m)=(s+\mathrm{par}(m)+1)(n-1)+1. \end{aligned}$$

Proof

From Theorem 6, one can check that $P_n$ is $C_m$-good. For the case m is even, take $a_1=a_2=m/2$ and apply Theorems 3, 2 and 4; for the case m is odd, apply Theorems 2 and 4. In both case, we have the assertion. $\square$

The wheel $W_m$ is the graph obtained by joining $K_1$ and a cycle $C_m$. For the case $(s,t)=(1,0)$ in Corollary 3, we can get the values of path-wheel Ramsey numbers $R(P_n,C_m)$ under the condition of Corollary 3, which was already obtained by Chen et al. [6].

Theorem 7

If $n\ge 2$, then

$$\begin{aligned}R(P_n,mK_1)=m.\end{aligned}$$

This theorem is trivial and the following corollary can be get immediately. We omit the proof.

Corollary 4

Let $t\ge 0$ be an integer. If $m\le \lceil n/2\rceil$, then

$$\begin{aligned}R(P_n,L_s^t\vee mK_1)=s(n-1)+1.\end{aligned}$$

For $m\ge 2$, the graph $K_{1,m}$ is called a star; the graph $K_2\vee mK_1$ is called a book; and the graph $K_t\vee mK_1$, $t\ge 3$, is called a generalized book. We remark here that the Ramsey numbers of paths versus stars and paths versus (generalized) books under the condition of Corollary 4 was already obtained by Parsons [12], and Rousseau and Sheehan [14], respectively.

Theorem 8

(Faudree and Schelp [8]) If $n,m_i\ge 2$, $1\le i\le k$, then

$$\begin{aligned} R\left( P_n,\bigcup _{i=1}^kP_{m_i}\right) =\max \left\{ n+\sum _{i=1}^k\left\lfloor \frac{m_i}{2}\right\rfloor -1, \sum _{i=1}^km_i+\left\lfloor \frac{n}{2}\right\rfloor -1\right\} . \end{aligned}$$

Corollary 5

Let $t\ge 0$ be an integer. If $m_i\ge 2$, $1\le i\le k$ and $\sum _{i=1}^k\lceil m_i/2\rceil \le \lceil n/2\rceil$, then

$$\begin{aligned} R\left( P_n,L_s^t\vee \bigcup _{i=1}^kP_{m_i}\right) =(s+1)(n-1)+1. \end{aligned}$$

Proof

By Theorem 8, $P_n$ is $(\bigcup _{i=1}^kP_{m_i})$-good. Take

$$\begin{aligned} a_1=\sum _{i=1}^k\left\lceil \frac{m_i}{2}\right\rceil \text{ and } a_2=\sum _{i=1}^k\left\lfloor \frac{m_i}{2}\right\rfloor . \end{aligned}$$

By Theorem 3, $P_n$ is $(K_1\vee \bigcup _{i=1}^kP_{m_i})$-good. By Theorems 2 and 4 we get the assertion. $\square$

The graph $F_m=K_1\vee mK_2$ is called a fan. From the above corollary, we can see that if $m\le \lceil n/2\rceil$, then $R(P_n,F_m)=2n-1$. This result was already obtained by Saleman and Broersma [15].

Let $P_n^k$ be the k-th power of $P_n$, i.e., the graph with vertex set $\{v_1,\ldots ,v_n\}$ and edge set $\{v_iv_j: |i-j|\le k\}$.

Theorem 9

(Pokrovskiy [13]) If $n\ge k+1$, then

$$\begin{aligned} R(P_n,P_n^k)=k(n-1)+\left\lfloor \frac{n}{k+1}\right\rfloor . \end{aligned}$$

Corollary 6

Let $t\ge 0$ be an integer. If $n\ge k+1$, then

$$\begin{aligned} R(P_n,L_s^t\vee P_n^k)=(t+k)(n-1)+1. \end{aligned}$$

Proof

Note that $\chi (P_n^k)=k+1$ and $\sigma (P_n^k)=\lfloor n/(k+1)\rfloor$. By Theorem 9, $P_n$ is $P_n^k$-good. Take

$$\begin{aligned} a_i=\left\lfloor \frac{n+i-1}{k+1}\right\rfloor , 1\le i\le k+1. \end{aligned}$$

It is easy to see that $P_n^k$ is a subgraph of $K_{a_1,a_2,\ldots ,a_{k+1}}$. By Theorems 3, 2 and 4, we have the assertion. $\square$

Theorem 10

(Sudarsana et al. [18]) If $m\ge 2$, then

$$\begin{aligned} R(P_n,2K_m)=(m-1)(n-1)+2. \end{aligned}$$

Corollary 7

Let $t\ge 0$ be an integer. If $m\ge 2$ and $n\ge 3$, then

$$\begin{aligned} R(P_n,L_s^t\vee 2K_m)=(s+m-1)(n-1)+1. \end{aligned}$$

Proof

By Theorem 10, $P_n$ is $2K_m$-good. Take $a_i=2$, $1\le i\le m$. Note that $2K_m$ is a subgraph of $K_{a_1,a_2,\ldots ,a_m}$. By Theorems 3, 2 and 4, we have the assertion. $\square$

Theorem 11

(Sudarsana [17]) If $m,k\ge 2$ and $n\ge (k-2)((km-2)(m-1)+1)+3$, then

$$\begin{aligned} R(P_n,kK_m)=(m-1)(n-1)+k. \end{aligned}$$

Corollary 8

Let $t\ge 0$ be an integer. If $m,k\ge 2$ and $n\ge (k-2)((km-2)(m-1)+1)+3$, then

$$\begin{aligned} R(P_n,L_s^t\vee kK_m)=(s+m-1)(n-1)+1. \end{aligned}$$

Proof

By Theorem 11, $P_n$ is $kK_m$-good. Take $a_i=k$, $1\le i\le m$. Note that $kK_m$ is a subgraph of $K_{a_1,a_2,\ldots ,a_m}$. By Theorems 3, 2 and 4, we have the assertion. $\square$

The cocktail party graph (or hyperoctahedral graph) $H_m$ is the graph obtained by removing a perfect matching from a complete graph $K_{2m}$ (i.e., $H_m=\overline{mK_2}$).

Theorem 12

(Ali et al. [1]) If $n,m\ge 3$, then

$$\begin{aligned} R(P_n,H_m)=(n-1)(m-1)+2. \end{aligned}$$

Corollary 9

Let $t\ge 0$ be an integer. If $n,m\ge 3$, then

$$\begin{aligned} R(P_n,L_s^t\vee H_m)=(s+m-1)(n-1)+1. \end{aligned}$$

Proof

By Theorem 12, $P_n$ is $H_m$-good. Take $a_i=2$, $1\le i\le m$. Note that $H_m=K_{a_1,a_2,\ldots ,a_m}$. By Theorems 3, 2 and 4, we have the assertion. $\square$

The sunflower graph $SF_m$ is the graph on $2m+1$ vertices obtained by taking a wheel $W_m$ with hub x, an m-cycle $v_1v_2\cdots v_mv_1$, and additional m vertices $w_1,w_2,\ldots ,w_m$, where $w_i$ is joined by edges to $v_i,v_{i+1}$, $1\le i\le m$, where $v_{m+1}=v_1$.

Theorem 13

(Ali et al. [4]) If $m\ge 3$, then

$$\begin{aligned} R(P_n,SF_m)=\left\{ \begin{array}{ll} 2n+m/2-2, &{} m \text{ is } \text{ even } \text{ and } n\ge 4m^2-7m+4;\\ 3n-2, &{} m \text{ is } \text{ odd } \text{ and } n\ge 2m^2-9m+11. \end{array}\right. \end{aligned}$$

Corollary 10

Let $t\ge 0$ be an integer. If $m\ge 4$ is even and $n\ge 4m^2-7m+4$, or $m\ge 3$ is odd and $n\ge 2m^2-9m+11$, then

$$\begin{aligned} R(P_n,L_s^t\vee SF_m)=(s+2+\mathrm{par}(m))(n-1)+1. \end{aligned}$$

Proof

By Theorem 13, $P_n$ is $SF_m$-good. If m is even, then take $a_1=m+1$ and $a_2=a_3=m/2$; if m is odd, then $\sigma (SF_m)=1$. By Theorems 3, 2 and 4, we have the assertion. $\square$

The Beaded wheel $BW_m$ is a graph on $2m+1$ vertices which is obtained by inserting one vertex in each spoke of the wheel $W_m$.

Theorem 14

(Ali et al. [3]) If $m\ge 3$, then

$$\begin{aligned} R(P_n,BW_m)=\left\{ \begin{array}{ll} 2n-1 &{} m \text{ is } \text{ even } \text{ and } n\ge 2m^2-5m+4;\\ 2n &{} m \text{ is } \text{ odd } \text{ and } n\ge 2m^2-5m+3. \end{array}\right. \end{aligned}$$

Corollary 11

Let $t\ge 0$ be an integer. If $m\ge 4$ is even and $n\ge 2m^2-5m+4$, or $m\ge 3$ is odd and $n\ge 2m^2-5m+3$, then

$$\begin{aligned} R(P_n,L_s^t\vee BW_m)=(s+2)(n-1)+1. \end{aligned}$$

Proof

By Theorem 14, $P_n$ is $BW_m$-good. If m is even, then $\sigma (BW_m)=1$; if m is odd, then take $a_1=m$ and $a_2=a_3=(m+1)/2$. By Theorems 3, 2 and 4, we have the assertion. $\square$

The Jahangir graph $J_{2m}$ is a graph on $2m+1$ vertices consisting of a cycle $C_{2m}$ with one additional vertex which is adjacent alternatively to m vertices of $C_{2m}$.

Theorem 15

(Surahmat and Tomescu [19]) If $m\ge 2$ and $n\ge (4m-1)(m-1)+1$, then

$$\begin{aligned} R(P_n,J_{2m})=n+m-1. \end{aligned}$$

Corollary 12

Let $t\ge 0$ be an integer. If $m\ge 2$ and $n\ge (4m-1)(m-1)+1$, then

$$\begin{aligned} R(P_n,L_s^t\vee J_{2m})=(t+1)(n-1)+1. \end{aligned}$$

Proof

By Theorem 15, $P_n$ is $J_{2m}$-good. Take $a_1=m$ and $a_2=m+1$. By Theorems 3, 2 and 4, we have the assertion. $\square$

The generalized Jahangir graph $J_{k,m}$ is a graph on $km+1$ vertices consisting of a cycle $C_{km}$ with one additional vertex which is adjacent to m vertices of the $C_{km}$ each of which is at distance k to the next one on $C_{km}$.

Theorem 16

(Ali et al. [2]) If $m,k\ge 2$, then

$$\begin{aligned} R(P_n,J_{k,m})=\left\{ \begin{array}{ll} n+km/2-1, &{} k \text{ is } \text{ even } \text{ and } n\ge (2km-1)(km/2-1)+1;\\ 2n-1 &{} k \text{ is } \text{ odd, } m \text{ is } \text{ even } \text{ and } n\ge km(km-2)/2;\\ 2n &{} k,m \text{ are } \text{ odd } \text{ and } n\ge (km-1)^2/2. \end{array}\right. \end{aligned}$$

Corollary 13

Let $t\ge 0$ be an integer. If $n,m,k\ge 2$, and if k is even and $n\ge (2km-1)(km/2-1)+1$, or k is odd, m is even and $n\ge km(km-2)/2$, or k, m are odd and $n\ge (km-1)^2/2$, then

$$\begin{aligned} R(P_n,L_s^t\vee J_{k,m})=(s+1+\mathrm{par}(k))(n-1)+1. \end{aligned}$$

Proof

By Theorem 16, $P_n$ is $J_{k,m}$-good. If k is even, then take $a_1=km/2+1$ and $a_2=km/2$; if k is odd, then take

$$\begin{aligned} a_1=m\cdot \left\lfloor \frac{k+2}{3}\right\rfloor +1, a_2=m\cdot \left\lfloor \frac{k+1}{3}\right\rfloor \text{ and } a_3=m\cdot \left\lfloor \frac{k}{3}\right\rfloor . \end{aligned}$$

By Theorems 3, 2 and 4, we have the assertion. $\square$

3 Proof of Theorem 3

From Theorem 1, it is sufficient to prove that $R(P_n,K_1\vee H)\le k(n-1)+1$. Let G be a graph of order $k(n-1)+1$. Suppose that G contains no $P_n$ and $\overline{G}$ contains no $K_1\vee H$.

Since H is a subgraph of $K_{a_1,a_2,\ldots ,a_k}$, we have

$$\begin{aligned} \sigma (H)\le a_k\le \left\lceil \frac{k(n-1)+1}{2k}\right\rceil =\left\lceil \frac{n}{2}-\frac{k-1}{2k}\right\rceil =\left\lceil \frac{n}{2}\right\rceil . \end{aligned}$$

Since $P_n$ is H-good,

$$\begin{aligned} R(P_n,H)=(\chi (H)-1)(n-1)+\sigma (H)\le (k-1)(n-1)+\left\lceil \frac{n}{2}\right\rceil . \end{aligned}$$

If there is a vertex v in G with $d(v)\le \lfloor n/2\rfloor -1$, then let $G'$ be a subgraph of G induced by $V(G)-\{v\}-N(v)$, where N(v) is the set of vertices adjacent to v in G. Note that

$$\begin{aligned} \nu (G')= & {} \nu (G)-1-d(v)\ge k(n-1)+1-\left\lfloor \frac{n}{2}\right\rfloor \\= & {} (k-1)(n-1)+\left\lceil \frac{n}{2}\right\rceil \ge R(P_n,H). \end{aligned}$$

This implies that $G'$ contains a path $P_n$ or $\overline{G'}$ contains a subgraph isomorphic to H. Note that v is nonadjacent to every vertex of $G'$. G contains a $P_n$ or $\overline{G}$ contains a $K_1\vee H$, a contradiction. Thus we assume that $\delta (G)\ge \lfloor n/2\rfloor$.

If there is a component B of G with $\nu (B)\ge n$, then by Dirac’s Theorem (see [7]), B contains a $P_n$, a contradiction. Thus we assume that every component of G has order at most $n-1$. Note that the minimum degree of G is at least $\lfloor n/2\rfloor$. Every component of G has order between $\lfloor n/2\rfloor +1$ and $n-1$.

If $\omega (G)\le k$, then $\nu (G)\le k(n-1)$; and if $\omega (G)\ge 2k$, then $\nu (G)\ge k(n+1)$, both a contradiction. This implies that

$$\begin{aligned} k+1\le \omega (G)\le 2k-1. \end{aligned}$$

Let $\mathcal {B}=\{B_1,B_2,\ldots ,B_\omega \}$, $\omega =\omega (G)$, be the set of the components of G. We assume without loss of generality that $\nu (B_1)\ge \nu (B_2)\ge \cdots \ge \nu (B_\omega )$. Thus we have

$$\begin{aligned} \nu (B_i)\ge \left\lceil \frac{\nu (G)-(i-1)(n-1)}{\omega -i+1}\right\rceil =\left\lceil \frac{(k-i+1)(n-1)+1}{\omega -i+1}\right\rceil , 1\le i\le k< \omega . \end{aligned}$$

Now we partition $\mathcal {B}$ into $k+1$ parts such that the order sum of the components in the ith part is at least $a_i$, $1\le i\le k$.

Let $t=\omega -k-1$. For $1\le i\le t$, let $\mathcal {B}_i=\{B_{\omega -2i+1},B_{\omega -2i}\}$; for $t+1\le i\le k$, let $\mathcal {B}_i=\{B_{i-t}\}$; and let $\mathcal {B}_{k+1}=\{B_\omega \}$.

If $1\le i\le t$, then $\mathcal {B}_i$ contains two components each of which has order at least $\lfloor n/2\rfloor +1$. Thus $\sum \{\nu (B_j): B_j\in \mathcal {B}_i\}\ge n+1$. On the other hand,

$$\begin{aligned} a_i\le \left\lceil \frac{k(n-1)+1}{k+i}\right\rceil \le \left\lceil \frac{k(n-1)+1}{k}\right\rceil =n<\sum _{B_j\in \mathcal {B}_i}\nu (B_j). \end{aligned}$$

If $t+1\le i\le k$, then $\mathcal {B}_i=\{B_{i-t}\}$. Note that

$$\begin{aligned} \nu (B_{i-t})\ge \left\lceil \frac{(k-i+t+1)(n-1)+1}{\omega -i+t+1}\right\rceil =\left\lceil \frac{(\omega -i)(n-1)+1}{2\omega -k-i}\right\rceil . \end{aligned}$$

Since $\omega -k\le i\le k$, one can check that

$$\begin{aligned} a_i\le \left\lceil \frac{k(n-1)+1}{k+i}\right\rceil \le \left\lceil \frac{(\omega -i)(n-1)+1}{2\omega -k-i}\right\rceil \le \nu (B_{i-t}). \end{aligned}$$

Clearly $\nu (B_\omega )\ge 1$. Thus $\overline{G}$ contains a $K_{a_1,a_2,\ldots ,a_k,1}$, which is a supergraph of $K_1\vee H$, our final contradiction.

The proof is complete.