Stars versus stripes Ramsey numbers

TL;DR

This paper calculates specific multicolor Ramsey numbers involving stars and matchings, extending previous results and providing new bounds for these combinatorial parameters.

Contribution

It generalizes and strengthens existing results on Ramsey numbers involving stars and matchings, offering explicit computations for complex graph configurations.

Findings

Computed Ramsey numbers for combinations of stars and matchings.

Extended known bounds and provided new exact values.

Generalized previous results by Cockayne, Lorimer, Gyárfás, and Sárközy.

Abstract

For given simple graphs $G_1, G_2, \ldots , G_t$, the Ramsey number $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ such that if the edges of the complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs $H_1,H_2,\ldots,H_t$, then at least one $H_i$ has a subgraph isomorphic to $G_i$. In this paper, for positive integers $t_1,t_2,\ldots, t_s$ and $n_1,n_2,\ldots, n_c$ the Ramsey number $R(S_{t_1}, S_{t_2},\ldots ,S_{t_s}, n_1K_2,n_2K_2,\ldots,n_cK_2)$ is computed, where $nK_2$ denotes a matching (stripe) of size $n$, i.e., $n$ pairwise disjoint edges and $S_{n}$ is a star with $n$ edges. This result generalizes and strengthens significantly a well-known result of Cockayne and Lorimer and also a known result of Gy\'{a}rf\'{a}s and S\'{a}rk\"{o}zy.

Equations

$$R(S_{t_1},S_{t_2},\dots ,S_{t_s},n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)=\max \{r+\delta ,n_1\}+\sum _{i=1}^c(n_i-1)+1,$$

$$R(S_t,n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)=\max \{t,n_1\}+\sum _{i=1}^c(n_i-1)+1.$$

$$R(S_t,n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)=R(n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)=n_1+\sum _{i=1}^c(n_i-1)+1,$$

$$G\to (n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2).$$

$$R(S_t,n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)=n.$$

$$f(t_{1},t_{2},\ldots,t_{s},n_{1},n_{2},\ldots,n_{c})=\left\{\begin{array}[]{ll}\max\{r,n_{1}\}+\Lambda+1&\Sigma~{}{and~{}at~{}least~{}one}~{}t_{i}~{}{is~{}even}~{}{and}\\ &~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Sigma<\max\{n_{1},2n_{2}\},\vspace{.6 cm}\\ \max\{r-1,n_{1}\}+\Lambda+1&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}otherwise.\end{array}\right.$$

$$G\to (n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2).$$

$${\mathrm{deg}}_G(u)+{\mathrm{deg}}_G(v)=\sum _{i=1}^c({\mathrm{deg}}^i(u)+{\mathrm{deg}}^i(v))\le 2\Lambda .(1)$$

$$\mathrm{deg}(u)=\mathrm{deg}(v)=\Lambda ,(2)$$

$${\mathrm{deg}}^i(u)+{\mathrm{deg}}^i(v)=2(n_i-1),i=1,2,\dots ,c(3)$$

$$R(S_{t_1},S_{t_2},\dots ,S_{t_s},n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)\le R(S_{t_1},S_{t_2},\dots ,S_{t_s},(\Lambda +1)K_2).$$

$$R(S_{t_1},S_{t_2},\dots ,S_{t_s},(\Lambda +1)K_2)=\Sigma +\Lambda +1,$$

$$R(S_{t_1},S_{t_2},\dots ,S_{t_s},n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)\le \Sigma +\Lambda +1.$$

$$G[V(G_1)\cup \{v\}]=G[V(G_2)\cup \{v\}]=K_{\Lambda +1}.$$

$$|G^{\prime }|=f(t_1,t_2,\dots ,t_s,m_1,m_2,\dots ,m_c)=\Sigma +{\Lambda }^{\prime }+1,$$

$$n_1=\sum _{i=2}^c(n_i-1)+k-1.(4)$$

$$f(t_1,t_2,\dots ,t_s,m_1,m_2,\dots ,m_c)=\Sigma +\sum _{i=1}^c(n_i-l_i-1)+1,$$

$$R(S_{t_1},S_{t_2},\dots ,S_{t_s},n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)\ge \max \{r,n_1\}+\Lambda +1.$$

$$R(S_{t_1},S_{t_2}\dots ,S_{t_s},n_1{K}_2,n_2{K}_2,\dots ,n_c{K}_2)\ge f(t_1,\dots ,t_s,n_1,\dots ,n_c).$$

Full text

1 Introduction

In this paper, we only concerned with undirected simple finite graphs and we follow [1] for terminology and notations not defined here. For a graph $G$, we denote its vertex set, edge set, minimum degree, maximum degree and complement graph by $V(G)$, $E(G)$, $\delta(G)$, $\Delta(G)$ and $\bar{G}$, respectively. If $v\in V(G)$, we use $\deg_{G}{(v)}$ and $N_{G}(v)$ $($or simply $\deg{(v)}$ and $N(v))$ to denote the degree and the neighbors of $v$ in $G$, respectively. Also, we use $nK_{2}$ to denote a matching (stripe) of size $n$, i.e., $n$ pairwise disjoint edges and as usual, a complete graph on $n$ vertices, a star with $n$ edges and a balanced complete bipartite graph on $2n$ vertices are denoted by $K_{n}$, $S_{n}$ and $K_{n,n}$, respectively. In addition, for disjoint subsets $A$ and $B$ of the vertex set of a graph $G$, we use $[A,B]$ to denote the bipartite subgraph of $G$ with partite sets $A$ and $B$.

If $G$ is a graph whose edges are colored by $c$ colors, we use $G^{i}$, $1\leq i\leq c$, to denote the subgraph of $G$ induced by the edges of the $i$-th color. Moreover, for a vertex $v$ of $G$, we use $\deg^{i}(v)$ and $N^{i}(v)$ to denote the degree and the neighbors of $v$ in $G^{i}$, respectively.

Recall that an edge coloring of $G$ is called proper if adjacent edges are assigned different colors. The minimum number of colors for a proper edge coloring of $G$ is called the chromatic index of $G$ and is denoted by $\chi^{\prime}(G)$. It is well known that for a bipartite graph $G$, we have $\chi^{\prime}(G)=\Delta(G)$, see [1].

Let $G,G_{1},G_{2},\ldots,G_{c}$ be given simple graphs. We write $G\rightarrow(G_{1},G_{2},\ldots,G_{c})$, if the edges of $G$ are partitioned into $c$ disjoint color classes giving $c$ graphs $H_{1},H_{2},\ldots,H_{c}$, then at least one $H_{i}$ has a subgraph isomorphic to $G_{i}$. For given simple graphs $G_{1},G_{2},\ldots,G_{c}$, the multicolor Ramsey number $R(G_{1},G_{2},\ldots,G_{c})$ is defined as the smallest positive integer $n$ such that $K_{n}\rightarrow(G_{1},G_{2},\ldots,G_{c})$. The existence of such a positive integer is guaranteed by the Ramsey’s classical result [8]. For a survey on Ramsey theory, we refer the reader to the regularly updated survey by Radziszowski [7].

There is very little known about $R(G_{1},G_{2},\ldots,G_{c})$ for $c\geq 3$, even for very special graphs. In this paper, we consider the case that $G_{i}$’s are stars or stripes. The Ramsey number of stars or stripes were investigated by several authors. The Ramsey number of stars is determined by Burr and Roberts [2] and the Ramsey number for stripes was determined by Cockayne and Lorimer [3]. In fact they showed that $R(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{c})=n_{1}+\sum_{i=1}^{c}(n_{i}-1)+1$ for $n_{1}\geq n_{2}\geq\cdots\geq n_{c}$. In [6] Gyárfás and Sárközy determined the exact value of the Ramsey number of a star versus two stripes and then they used this result to give a positive answer to a conjecture of Schelp in an asymptotic sense. It is also worth noting that the Ramsey number for many stars and one stripe was determined in [4] as follows.

Theorem 1.1**.**

[4]* Let $t_{1},t_{2},\ldots,t_{s}$ be positive integers, $\Sigma=\sum_{i=1}^{s}(t_{i}-1)$ and $n\geq 1$. Then

1) $R(S_{t_{1}},S_{t_{2}},\ldots,S_{t_{s}},nK_{2})=2n,$ if $\Sigma<n$,

2) $R(S_{t_{1}},S_{t_{2}},\ldots,S_{t_{s}},nK_{2})=\Sigma+n,$ if $\Sigma\geq n$, $\Sigma$ is even and some $t_{i}$ is even,

3) $R(S_{t_{1}},S_{t_{2}},\ldots,S_{t_{s}},nK_{2})=\Sigma+n+1,$ otherwise.

Note that, using Theorem 1.1 for $n=1$, we conclude that $R(S_{t_{1}},S_{t_{2}},\ldots,S_{t_{s}})=\Sigma+1,$ if $\Sigma$ and at least one $t_{i}$ are even and $R(S_{t_{1}},S_{t_{2}},\ldots,S_{t_{s}})=\Sigma+2,$ otherwise.

The aim of this paper is the following theorem which provides the exact value of the Ramsey number of any number of stars versus any number of stripes. This theorem extends known results on the Ramsey number of stars and stripes in the literature.

Theorem 1.2**.**

Let $t_{1},t_{2},\ldots,t_{s}$ and $n_{1},n_{2},\ldots,n_{c}$ be positive integers, $\Sigma=\sum_{i=1}^{s}(t_{i}-1)$ and $r=R(S_{t_{1}},S_{t_{2}},\ldots,S_{t_{s}})$. If $n_{1}\geq n_{2}\geq\cdots\geq n_{c}$, then

[TABLE]

where $\delta=0$ if $\Sigma<\max\{n_{1},2n_{2}\}$, $\Sigma$ is even and some $t_{i}$ is even, and $\delta=-1$, otherwise.

As an easy corollary of Theorem 1.2, we have the following result which generalizes a known result of Gyárfás and Sárközy [6] on the Ramsey number of one star versus two stripes.

Corollary 1.3**.**

Let $t\geq 1$ and $n_{1}\geq n_{2}\geq\cdots\geq n_{c}$ be positive integers. Then

[TABLE]

By Corollary 1.3, for $t\leq n_{1}$, $n_{1}=\max\{n_{1},n_{2},\ldots,n_{c}\}$, we have

[TABLE]

which strengthens significantly a well-known result of Cockayne and Lorimer on the Ramsey number of stripes. In the other word, if $G$ is a graph obtained by deleting the edges of a graph with maximum degree $(n_{1}-1)$ from a complete graph on $R(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$ vertices, then

[TABLE]

In addition, we obtain the following interesting result if we investigate to Corollary 1.3, when $t\geq n_{1}$.

Corollary 1.4**.**

Let $n_{1}\geq n_{2}\geq\cdots\geq n_{c}$ be arbitrary positive integers, and let $G$ be a graph on $n\geq n_{1}+\sum_{i=1}^{c}(n_{i}-1)+1$ vertices such that $\delta(G)\geq\sum_{i=1}^{c}(n_{i}-1)+1$. Then

[TABLE]

**Proof. **Set $t=n-\sum_{i=1}^{c}(n_{i}-1)-1$. Clearly $t\geq n_{1}$ and so by Corollary 1.3, we have

[TABLE]

Since $G$ has $n$ vertices and $\delta(G)\geq\sum_{i=1}^{c}(n_{i}-1)+1$, we have $\Delta(\bar{G})\leq t-1$, which means that $\bar{G}$ is a $S_{t}$-free graph and so the assertion holds by the above equation. $\hfill\blacksquare$

It is also worth noting that the condition on the minimum degree in Corollary 1.4 is best possible. Indeed, let $G$ be a graph on $n\geq R(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$ vertices whose vertex set is partitioned into disjoint sets $A$, $B$ with $|A|=\Lambda=\sum_{i=1}^{c}(n_{i}-1)$, $|B|=n-\Lambda$ and let $E(G)=\{uv|\{u,v\}\cap A\neq\emptyset\}$. Now, set $V_{0}=B$ and consider a partition of vertices of $A$ into sets $V_{1},V_{2},\ldots,V_{c}$ of sizes $n_{1}-1$, $n_{2}-1$, $\ldots,n_{c}-1$, respectively. For each $i=1,2,\ldots,c$ color with the $i$-th color all edges within $V_{i}$ or edges with one vertex in $V_{i}$ and one in $V_{j}$, where $j<i$. In this coloring, the largest monochromatic matching of color $i$ has $n_{i}-1$ edges, while the minimum degree of $G$ is $\Lambda$.

2 Proof of Theorem 1.2

In order to prove Theorem 1.2, we need some lemmas. First, we start with the following simple but useful lemma.

Lemma 2.1**.**

Let $t_{1},t_{2},\ldots,t_{s}$ be positive integers, $\Sigma=\sum_{i=1}^{s}(t_{i}-1)$ and let $H$ be a graph with $\chi^{\prime}(H)\leq\Sigma$. Then $E(H)$ can be decomposed into edge-disjoint subgraphs $H_{1},H_{2},\ldots,H_{s}$ such that $\Delta(H_{i})\leq t_{i}-1.$

**Proof. ** Consider a proper edge-coloring of $H$ with $\chi^{\prime}(H)$ colors. Partition the set of colors into $s$ sets $A_{1},A_{2},\ldots,A_{s}$ of sizes at most $t_{1}-1,t_{2}-1,\ldots,t_{s}-1$, respectively. Let $H_{i}$, $1\leq i\leq s$, be the subgraph of $H$ induced by the edges of colors in $A_{i}$. Clearly $H_{i}$’s are the desired subgraphs which decompose $E(H)$. $\hfill\blacksquare$

An alternating cycle in an edge colored graph is a cycle which is properly colored i.e. no two consecutive edges in the cycle have the same color. We say that a vertex $v$ in an edge colored graph $G$ separates colors if no component of $G-v$ is joined to $v$ by at least two edges of different colors. Grossman and Häggkvist gave a sufficient condition under which a two-edge colored graph must have an alternating cycle. In [5] Grossman and Häggkvist proved that if $G$ is a graph whose edges are colored red and blue and there is no alternating cycle in $G$, then $G$ contains a vertex $v$ that separates the colors. Bang-Jensen and G. Gutin asked whether Grossman and Häggkvist’s result could be extended to edge-colored graphs in general, where there is no constraint on the number of colors. In [9] Yeo gave an affirmative answer to this question as follows.

Theorem 2.2**.**

([9]) If $G$ is a $c$-edge-colored graph, $c\geq 2$, with no alternating cycle, then there is a vertex $v\in V(G)$ such that no connected component of $G-v$ is joined to $v$ with edges of more than one color, i.e $G$ contains a vertex separating colors.

Let $t_{1},t_{2},\ldots,t_{s}$ be positive integers, $\Sigma=\sum_{i=1}^{s}(t_{i}-1)$ and $r=R(S_{t_{1}},S_{t_{2}},\ldots,S_{t_{s}})$. Also let $n_{1}\geq n_{2}\geq\cdots\geq n_{c}$ be positive integers and $\Lambda=\sum_{i=1}^{c}(n_{i}-1)$. Set

[TABLE]

In fact, $f(t_{1},t_{2},\ldots,t_{s},n_{1},n_{2},\ldots,n_{c})$ is the number that we claimed is equal to the Ramsey number $R(S_{t_{1}},S_{t_{1}},\ldots,S_{t_{s}},n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$ in Theorem 1.2. Using these notations, we have the following lemma.

Lemma 2.3**.**

Let $t_{1},t_{2},\ldots,t_{s}$ and $n_{1},n_{2},\ldots,n_{c}$ with $n_{1}\geq n_{2}\geq\cdots\geq n_{c}$ be positive integers and let $G$ be a graph on $f(t_{1},t_{2},\ldots,t_{s},n_{1},n_{2},\ldots,n_{c})$ vertices such that $\bar{G}\nrightarrow(S_{t_{1}},S_{t_{1}},\ldots,S_{t_{s}}).$ Then

[TABLE]

**Proof. **Assume that the statement of this lemma is not correct and suppose that a counterexample exists. Therefore, there are some positive integers $t_{1},t_{2},\ldots,t_{s}$ and $n_{1},n_{2},\ldots,n_{c}$ with $n_{1}=\max\{n_{1},n_{2},\ldots,n_{c}\}$, and a graph $G$ on $f(t_{1},t_{2},\ldots,t_{s},n_{1},n_{2},\ldots,n_{c})$ vertices, such that $\bar{G}\nrightarrow(S_{t_{1}},S_{t_{1}},\ldots,S_{t_{s}})$ and $G\nrightarrow(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2}).$ Note that $c\geq 2$, by Theorem 1.1.

Among all counterexamples let $G$ be a minimal one having the maximum possible number of edges, i.e. $G$ is a graph satisfies the following conditions:

(a) The number of vertices of $G$, $f(t_{1},t_{2},\ldots,t_{s},n_{1},n_{2},\ldots,n_{c})$, is as small as possible.

(b) Among all counterexamples satisfying (a), $G$ is a counterexample with minimum $c$, i.e. no counterexample is colored with less than $c$ colors.

(c) Among all counterexamples satisfying (a) and (b), $G$ is one having the maximum possible number of edges.

The fact $G\nrightarrow(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$ implies that the edges of $G$ can be colored by colors $\beta_{1},\beta_{2},\ldots,\beta_{c}$ so that for each $i$, $1\leq i\leq c$, the induced graph on edges of color $\beta_{i}$ does not contain a subgraph isomorphic to $n_{i}K_{2}$. Let $G^{i}$ be the subgraph of $G$ induced by the edges of color $\beta_{i}$. As $|V(G)|\geq R(n_{1}k_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$, we deduce that $G$ is not a complete graph. Let $u,v$ be non-adjacent vertices in $G$. As $G$ satisfies (a), (b) and (c), $n_{i}K_{2}\subseteq G^{i}+uv$ (to see this, it only suffices to add the edge $uv$ to $G$ and color $uv$ by $\beta_{i}$ and then use the property (c) of $G$) which means that $(n_{i}-1)K_{2}\subseteq G^{i}$. Let $M$ be the matching of size $(n_{i}-1)$ in $G^{i}$. Since $n_{i}K_{2}\nsubseteq G^{i}$, we must have $N^{i}(u),N^{i}(v)\subseteq V(M)$. Moreover, the fact $n_{i}K_{2}\nsubseteq G^{i}$ implies that for each edge $xy\in M$, the number of edges of color $i$ between $\{x,y\}$ and $\{u,v\}$ is at most 2. Thus $\deg^{i}(u)+\deg^{i}(v)\leq 2(n_{i}-1)$, for each $i=1,2,\ldots,c$. Therefore,

[TABLE]

Since $\bar{G}\nrightarrow(S_{t_{1}},S_{t_{1}},\ldots,S_{t_{s}})$, there is a $s$ coloring of edges of $\bar{G}$ such that the graph induced by the $i$-th color does not contain $S_{t_{i}}$ as a subgraph. Thus, for every vertex $v\in V(G)$, we have $\deg_{\bar{G}}(v)\leq\Sigma$. (Indeed, if $v$ is a vertex with $\deg_{\bar{G}}(v)\geq\Sigma+1$, then the Pigeonhole principle implies that any $s$ coloring of the edges of $\bar{G}$ contains a monochromatic $S_{t_{i}}$ of $i$-th color with center $v$, for some $i$, a contradiction). Therefore, $\delta(G)\geq f(t_{1},t_{2},\ldots,t_{s},n_{1},n_{2},\ldots,n_{c})-\Sigma-1$. An easy calculation shows that $\delta(G)\geq\Lambda+1$ unless $\Sigma\geq\max\{n_{1},2n_{2}\}$, $\Sigma$ is even and some $t_{i}$ is even and in this case, we have $\delta(G)\geq\Lambda$. If $\delta(G)\geq\Lambda+1$, then for every pair of vertices $u,v$, $\deg_{G}(u)+\deg_{G}(v)\geq 2(\Lambda+1)$. Using (1), we deduce that a counterexample could not exist unless $\Sigma\geq\max\{n_{1},2n_{2}\}$, $\Sigma$ is even and some $t_{i}$ is even. Therefore, hereafter we may suppose that $\Sigma$ and at least one $t_{i}$ is even and $\Sigma\geq\max\{n_{1},2n_{2}\}$. Note that, in this case we have $f(t_{1},t_{2},\ldots,t_{s},n_{1},n_{2},\ldots,n_{c})=\Sigma+\Lambda+1$. By (1) and the fact $\delta(G)\geq\Lambda$ we conclude that for every pair of non-adjacent vertices $u,v$ in $G$:

[TABLE]

[TABLE]

Claim 1. $\Sigma\leq\Lambda$.

*Proof of Claim 1. *On the contrary, let $\Sigma\geq\Lambda+1$. It is easy to see that

[TABLE]

As $\Sigma$ and some $t_{i}$ are even, by Theorem 1.1 we have

[TABLE]

which implies that

[TABLE]

This means that $K_{N}\rightarrow(S_{t_{1}},S_{t_{2}},\ldots,S_{t_{s}},n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$, where $N=\Sigma+\Lambda+1.$ Therefore $\bar{G}\rightarrow(S_{t_{1}},S_{t_{1}},\ldots,S_{t_{s}})$ or $G\rightarrow(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$, a contradiction. $\hfill\square$

Claim 2. $G$ is a 2-connected graph.

Proof of Claim 2. By Claim 1, one can easily check that $\delta(G)\geq\Lambda\geq\frac{|V(G)|}{2}$, unless $\Sigma=\Lambda$. Therefore, by the Dirac’s Theorem [1], $G$ is a hamiltonian graph and so a 2-connected graph unless $\Sigma=\Lambda$. Now, assume that $\Sigma=\Lambda$. In this case, $|V(G)|=2\Lambda+1$ and $\delta(G)\geq\Lambda$. Clearly, $G$ is connected (in fact the diameter of $G$ is two, since every two non-adjacent vertices have a common neighbor). If there is a cut vertex $v$ of $G$, then $G-v$ has exactly two components $G_{1},G_{2}$ with

[TABLE]

Now, we claim that all edges of $G_{1}\cup\{v\}$ (also $G_{2}\cup\{v\}$) have the same color. To see this, let $v_{1}$ be an arbitrary vertex of $G_{1}$ and let the edge $vv_{1}$ is colored by $\beta_{i}$, for some $i$, $1\leq i\leq c$. Since $G_{1}\cup\{v\}$ is a complete graph and $v_{1}$ is an arbitrary vertex of $G_{1}$, in order to show that all edges of $G_{1}\cup\{v\}$ have the same color, it only suffices to show that all edges of $G_{1}\cup\{v\}$ incident to $v_{1}$ are of color $\beta_{i}$. On the contrary, assume that the edge $v_{1}v_{2}$ of $G_{1}\cup\{v\}$ is of color $\beta_{j}$, where $j\neq i$. Now let $M_{1},M_{2}$ be arbitrary perfect matchings in $G_{1},G_{2}$, respectively, where $v_{1}v_{2}\in M_{1}$. Therefore, $|M_{1}\cup M_{2}|=\Lambda$ and we may assume that for each $t=1,2,\ldots,c$, the matching $M_{1}\cup M_{2}$ contains exactly $n_{t}-1$ edges of color $\beta_{t}$, since otherwise for some $1\leq i\leq c$, $G$ has a monochromatic matching of size $n_{i}$ with color $\beta_{i}$, which is impossible. Set $M=(M_{1}\setminus\{v_{1}v_{2}\})\cup M_{2}\cup\{vv_{1}\}$. Clearly $M$ contains a monochromatic matching of size $n_{i}$ with color $\beta_{i}$, which is again impossible. By a similar argument, all edges of $G_{2}\cup\{v\}$ have the same color. Therefore at most two colors are appeared on the edges of $G$, say $\beta_{i}$ and $\beta_{j}$ (for some $i$ and $j$). Without any loss of generality, we may assume that all edges within $G_{2}$ are of color $\beta_{j}$ and $j\neq 1$. As $\Lambda=\Sigma\geq\max\{n_{1},2n_{2}\}$ and $|V(G_{2})|=\Lambda$, we obtain that $|V(G_{2})|\geq 2n_{2}\geq 2n_{j}$ which means that $G_{2}$ contains a subgraph isomorphic to $n_{j}K_{2}$ of color $\beta_{j}$, a contradiction. $\hfill\square$

Now the analysis depends on the study of certain cycles in $G$. These are alternating cycles, colored with some colors $\beta\in\{\beta_{1},\beta_{2},\ldots,\beta_{c}\}$, having no two adjacent edges of the same color. The rest of the proof is devoted to prove that an alternating cycle exists in $G$.

Claim 3. $G$ has an alternating cycle.

*Proof of Claim 3. *On the contrary, assume that $G$ does not have an alternating cycle. Thus using Theorem 2.2, $G$ has a vertex $v$ separating colors. Since $G$ is 2-connected by Claim 2, all edges of $G$ incident to $v$ have the same color, say $\beta_{i}$. Set $G^{\prime}=G\setminus\{v\}$. Note that

[TABLE]

where $\Lambda^{\prime}=\Lambda-1$, and $m_{1},m_{2},\ldots,m_{c}$ are the numbers $n_{1},n_{2},\ldots,n_{i}-1,\ldots,n_{c}$ in the decreasing order. Clearly any $s$-coloring of the edges of $\bar{G}$ induces an $s$-coloring of the edges of $\bar{G^{\prime}}$. Therefore $\bar{G^{\prime}}\nrightarrow(S_{t_{1}},S_{t_{1}},\ldots,S_{t_{s}}).$ From the minimality of $G$ we deduce that $G^{\prime}$ has a subgraph $M$ isomorphic to $(n_{i}-1)K_{2}$ whose edges are colored by $\beta_{i}$. If the degree of $v$ as a separator vertex is at least $2n_{i}-1$, then there is a vertex $u\in N_{G}(v)$ which is unsaturated by the vertices of the matching $M$. Thus adding the edge $uv$ to the matching $M$ yields a monochromatic copy of $n_{i}K_{2}$ with color $\beta_{i}$ in $G$, which is impossible. Therefore, the proof of the claim will be completed if we prove that the degree of $v$ as a separator vertex is at least $2n_{i}-1$.

First let all edges of $G$ incident to $v$ have color $\beta_{i}$ and $i\geq 2$. Since $\Lambda\geq\Sigma$ by Claim 1, and also $\deg_{G}(v)\geq\delta(G)\geq\Lambda$, we obtain that $\deg_{G}(v)\geq\Sigma\geq\max\{n_{1},2n_{2}\}\geq 2n_{2}\geq 2n_{i}$ and we are done.

Now, let all edges of $G$ incident to $v$ have color $\beta_{1}$. Let $\deg_{G}(v)=2n_{1}-k$, for some $k\geq 2$. Note that the fact $\Sigma\geq\max\{n_{1},2n_{2}\}$ implies that $|V(G)|=\Sigma+\Lambda+1>2n_{1}$ and so $v$ is not adjacent to all vertices of $G$. Therefore, by (2) we obtain that $\deg_{G}(v)=2n_{1}-k=\Lambda$. This means that

[TABLE]

Since $\deg_{G}(v)=\Lambda$ and $|V(G)|=\Sigma+\Lambda+1$, the vertex $v$ has exactly $\Sigma$ non-neighbors in $G$. Let $S$ be the set of non-neighbors of $v$ in $G$. By (3), for every vertex $z\in S$, $\deg^{1}(z)=k-2$ and for $i=2,3,\ldots,c$ we have $\deg^{i}(z)=2n_{i}-2$. Since for every vertex $z\in S$, $\deg^{2}(z)=2n_{2}-2$, Equation (3) implies that the graph induced by the vertices of $S$ is a complete graph. Now, we prove that $G[S]$ contains an alternating cycle. By Theorem 2.2, $G[S]$ contains an alternating cycle unless there is a vertex which separates colors. Let $z$ be a vertex of $G[S]$ separating colors and all edges of $G[S]$ incident to $v$ have the same color, say $\beta_{i}$. If $i=1$ then $k-2\geq\deg^{1}_{G[S]}(z)\geq\Sigma-1$ which implies that $k-1\geq\Sigma\geq n_{1}$, which contradicts (4). If $i\geq 2$ then $2n_{i}-2\geq\deg^{i}_{G[S]}(z)\geq\Sigma-1$ which implies that $2n_{i}-1\geq\Sigma\geq 2n_{2}\geq 2n_{i}$, a contradiction. This contradiction shows that if $v$ is a vertex of $G$ separating colors and all edges of $G$ incident to $v$ have the same color $\beta_{i}$, then the degree of $v$ as a separator vertex is at least $2n_{i}-1$, which completes the proof of the Claim 3. $\hfill\square$

Now let $C$ be an alternating cycle of $G$ which has $l_{i}$ edges colored by $\beta_{i}$, for each $i=1,2,\ldots,c$, then it has $\sum_{i=1}^{c}l_{i}$ vertices. For each $i$, the $l_{i}$ edges of $C$ colored by $\beta_{i}$ form a subgraph isomorphic to $l_{i}K_{2}$. If $G^{\prime}=V(G)\setminus V(C)$, then the number of vertices in $G^{\prime}$ is

[TABLE]

where $m_{1},m_{2},\ldots,m_{c}$ are the numbers $n_{i}-l_{i}$ for $i=1,\ldots,c$ in the decreasing order. As $n_{1}\geq m_{1}=\max\{m_{1},m_{2},\ldots,m_{c}\}$ and $C$ is a subgraph of $G$ which is properly colored, from the minimality of $G$ we deduce that $G^{\prime}$ has a monochromatic subgraph isomorphic to $(n_{i}-l_{i})K_{2}$ whose edges are colored by $\beta_{i}$, for some $1\leq i\leq c$. Combining this with a monochromatic subgraph $l_{i}K_{2}$ of color $\beta_{i}$ in $C$, we obtain a subgraph isomorphic to $n_{i}K_{2}$ with color $\beta_{i}$ in $G$, a contradiction. This contradiction shows that this lemma is true and so the proof is completed.

$\hfill\blacksquare$

Now, we are ready to give a proof for Theorem 1.2 which provides the exact value of the Ramsey number $R(S_{t_{1}},S_{t_{2}},\ldots,S_{t_{s}},n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$.

Proof of Theorem 1.2. To see that the Ramsey number can not be less than the claimed number, first consider the case that $\Sigma<\max\{n_{1},2n_{2}\}$, $\Sigma$ is even and some $t_{i}$ is even. Since $\Sigma$ and some $t_{i}$ are even, $r=\Sigma+1$ by Theorem 1.1. If $\Sigma<n_{1}$, then consider a partition of $n_{1}+\Lambda$ vertices into sets $V_{1},V_{2},\ldots,V_{c}$ of sizes $2n_{1}-1$, $n_{2}-1$, $\ldots,n_{c}-1$ respectively. Color with the first color all edges which are incident with two vertices of $V_{1}$ and for each $i=2,\ldots,c$ color with the $i$-th color all edges having two vertices in $V_{i}$ or one vertex in $V_{i}$ and one in $V_{j}$ where $j<i$. Clearly, for each $i=1,2,\ldots,c$, the graph induced by the edges of the $i$-th color does not contain a subgraph isomorphic to $n_{i}K_{2}$.

If $n_{1}\leq\Sigma<2n_{2}$, then partition $\Sigma+\Lambda+1$ vertices into sets $V_{1}=A\cup B$, $V_{2}=C\cup D$, $V_{3},\ldots,V_{c}$ with $|A|=|B|=\Sigma$, $|C|=2n_{2}-\Sigma-1$, $|D|=n_{1}-n_{2}$ and $|V_{i}|=n_{i}-1$, $3\leq i\leq c$. Color all edges contained in $B,D$ and edges in $[B,C],[D,C],[B,D],[A,D]$ by the first color $\beta_{1}$, all edges contained in $A,C$ and edges in $[A,C]$ by $\beta_{2}$. For each $i=3,4,\ldots,c$, color with $\beta_{i}$ all edges having two vertices in $V_{i}$ or one vertex in $V_{i}$ and one in $V_{j}$ where $j<i$. Clearly, for each $i=1,2,\ldots,c$, the graph induced by the edges of color $\beta_{i}$ does not contain a subgraph isomorphic to $n_{i}K_{2}$. The remaining uncolored edges are $[A,B]$ which form a copy of $K_{\Sigma,\Sigma}$. By Lemma 2.1, the edges of $K_{\Sigma,\Sigma}$ can be colored by $s$-colors $\alpha_{1},\alpha_{2},\ldots,\alpha_{s}$ such that the induced graph on edges of color $\alpha_{i}$, $1\leq i\leq s$, does not contain $S_{t_{i}}$ as a subgraph. This yields an edge coloring of the complete graph on $\max\{r,n_{1}\}+\Lambda$ vertices with $s+c$ colors $\alpha_{1},\alpha_{2},\ldots,\alpha_{s}$ and $\beta_{1},\beta_{2},\ldots,\beta_{c}$ such that the induced graph on edges of color $\alpha_{i}$, $1\leq i\leq s$, does not contain $S_{t_{i}}$ as a subgraph and for each $i=1,2,\ldots,c$, the induced graph on edges of color $\beta_{i}$ does not contain a subgraph isomorphic to $n_{i}K_{2}$. This observation shows that if $\Sigma<\max\{n_{1},2n_{2}\}$, $\Sigma$ is even and some $t_{i}$ is even, then

[TABLE]

Now assume that the case “$\Sigma<\max\{n_{1},2n_{2}\}$, $\Sigma$ is even and some $t_{i}$ is even” does not occur. Consider a partition of $n=\max\{r-1,n_{1}\}+\Lambda$ vertices into sets $V_{0},V_{1},V_{2},\ldots,V_{c}$ of sizes $\max\{r-1,n_{1}\},n_{1}-1,n_{2}-1,\ldots,n_{c}-1$ respectively. For each $i=1,2,\ldots,c$, color with $\beta_{i}$ all edges within $V_{i}$ or edges with one vertex in $V_{i}$ and one in $V_{j}$, where $j<i$. Now, if $r-1\leq n_{1}$, then $|V_{0}|=n_{1}$, and in this case color all edges within $V_{0}$ by $\beta_{1}$. In fact this is a $c$-edge coloring of $K_{n_{1}+\Lambda}$ that does not have a matching of size $n_{i}$ of color $\beta_{i}$, $1\leq i\leq c$. If $r-1>n_{1}$, then $|V_{0}|=r-1$ and so there is an edge coloring of $K_{r-1}$ with $s$ colors $\alpha_{1},\ldots,\alpha_{s}$ without a monochromatic copy of $S_{t_{i}}$ of color $\alpha_{i}$, $1\leq i\leq s$. This yields an $(s+c)$-edge coloring of $K_{n}$ that does not have a monochromatic star $S_{t_{i}}$ with color $\alpha_{i}$, $1\leq i\leq s$, and no monochromatic matching of size $n_{i}$ in color $\beta_{i}$, $1\leq i\leq c$. Therefore

[TABLE]

To prove the other direction, consider a complete graph on $N=f(t_{1},t_{2},\ldots,t_{s},n_{1},n_{2},\ldots,n_{c})$ vertices whose edges are arbitrarily colored by $s+c$ colors $\alpha_{1},\alpha_{2},\ldots,\alpha_{s}$ and $\beta_{1},\beta_{2},\ldots,\beta_{c}$. Let $G$ be the graph induced by all edges of color $\beta_{1},\beta_{2},\ldots,\beta_{c}$ in $K_{N}$. If for each $i$, $1\leq i\leq s$, the subgraph induced by the edges of color $\alpha_{i}$ in $K_{N}$ does not contain a copy of $S_{t_{i}}$, then $\bar{G}\nrightarrow(S_{t_{1}},S_{t_{1}},\ldots,S_{t_{s}})$ and so Lemma 2.3 implies that $G\rightarrow(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2}).$ This means that for some $i$, $1\leq i\leq c$, the subgraph of $K_{N}$ induced on the edges of color $\beta_{i}$ contains a subgraph isomorphic to $n_{i}K_{2}$, which completes the proof of the theorem. $\hfill\blacksquare$

Acknowledgment

The research of the first and second authors are partially carried out in the IPM-Isfahan Branch and in part supported respectively by grants No. 94050217 and No. 94050057, from IPM.