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Graphs and Combinatorics

, Volume 35, Issue 2, pp 559–567 | Cite as

On Some Three Color Ramsey Numbers for Paths, Cycles, Stripes and Stars

  • Farideh Khoeini
  • Tomasz DzidoEmail author
Open Access
Original Paper
  • 49 Downloads

Abstract

For given graphs \(G_{1}, G_{2},\ldots , G_{k}, k \ge 2\), the multicolor Ramsey number\(R(G_{1}, G_{2},\ldots , G_{k})\) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with k colors, then it contains a monochromatic copy of \(G_{i}\) in color i, for some \(1 \le i \le k\). The main result of the paper is a theorem which establishes the connection between the multicolor Ramsey number and the appropriate multicolor bipartite Ramsey number together with the ordinary Ramsey number. The remaining part of the paper consists of a number of corollaries which are derived from the main result and from known results for Ramsey numbers and bipartite Ramsey numbers. We provide some new exact values or generalize known results for multicolor Ramsey numbers of paths, cycles, stripes and stars versus other graphs.

Keywords

Ramsey number Bipartite Ramsey number Cycle Path Stripe Star 

Mathematics Subject Classification

05D10 05C55 

1 Introduction

All graphs in this paper are undirected, finite and simple. The union of two graphs G and H, denoted by \(G\cup H,\) is a graph with vertex set \(V(G) \cup V(H)\) and edge set \(E(G) \cup E(H)\). The join of two graphs G and H, denoted by \(G+H,\) is a graph with vertex set \(V(G) \cup V(H)\) and edge set \(E(G) \cup E(H) \cup \lbrace uv \vert u\in V(G), v\in V(H) \rbrace \). The union of k disjoint copies of the same graph G is denoted by kG. \({\overline{G}}\) stands for the complement of the graph G. We denote by G[U] the subgraph of G induced by the vertex set U. By \(P_n\) and \(C_n\) we denote the path and cycle on n vertices, respectively. For a 3-edge coloring (say blue, red and green) of a graph G, we denote by \(G^{b}\) (\(G^{r}\) and \(G^{g}\)) the subgraph induced by the edges of color blue (red and green, respectively).

For given graphs \(G_{1}, G_{2},\ldots , G_{k}, k \ge 2\), the multicolor Ramsey number\(R(G_{1}, G_{2},\ldots , G_{k})\) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with k colors, then it contains a monochromatic copy of \(G_{i}\) in color i, for some \(1 \le i \le k\). The existence of such a positive integer is guaranteed by Ramsey’s classical result [17]. Ramsey numbers are still in the main stream of investigations and there are no many results in the field of multicolor and even three-color Ramsey numbers. There are a lot of open cases (see [18]).

The bipartite Ramsey number \(b(G_1, \ldots , G_k)\) is the smallest positive integer b such that any coloring of the edges of \(K_{b,b}\) with k colors contains a monochromatic copy of bipartite \(G_i\) in the i-th color, for i, \(1 \le i \le k\).

In 2005, the second author began to determine the exact values for three color Ramsey numbers for two paths and one cycle. He proved that for \(n \ge 6\), \(R(P_3,P_3,C_n)=n\) and \(R(P_3,P_4,C_n)=n+1\) [4]. In 2006 Dzido et al. [6] proved that \(R(P_4,P_4,C_n)=n+2\) and \(R(P_3,P_5,C_n)=n+1\). In 2009 Dzido and Fidytek [5] (and independently Bielak in [2]) obtained the exact value of \(R(P_{i},P_{k},C_{m})\) for several values of ik and m.

Theorem 1

[2, 5] Let ikm be integers such that \(m \ge 3\) is odd, \(k \ge m,\) and \(k > \frac{3i^2-14i+25}{4}\) when i is odd,  and \(k > \frac{3i^2-10i+20}{8}\) when i is even. Then
$$\begin{aligned} R(P_i,P_k,C_m) = 2k + 2\Big \lfloor \frac{i}{2} \Big \rfloor - 3. \end{aligned}$$
In Sect. 3.1 we extend this result to other conditions on the length of paths and a cycle. By a short proof we show that for integers \(n_0, n_1, n_2\) such that \(n_0\) is sufficiently large, if \( n_{1}=2s\) and \(n_{2}=2m\) such that \(m-1<2s\), or \(n_{1}=n_{2}=2s\) or \(n_{1}=2s+1\) and \(n_{2}=2m\) such that \(s<m-1<2s+1\) then
$$\begin{aligned} R(C_{n_0}, P_{n_{1}},P_{n_{2}}) = n_0 + \Big \lfloor \frac{n_1}{2} \Big \rfloor + \Big \lfloor \frac{n_2}{2} \Big \rfloor -2. \end{aligned}$$
It should be noted that this result can be also obtained as a consequence of Theorem 2.2 in [14].
For the Ramsey number of paths a well-known theorem of Gerencsér and Gyárfás [10] states that \(R(P_{n},P_{m})=m+\lfloor n/2 \rfloor -1\) where \(m \ge n \ge 2\). In 1975 [7] Faudree and Schelp determined \(R(P_{n_1},P_{n_2},P_{n_3})\) for the case \(n_{1}\ge 6(n_{2}+n_{3})^{2}\) and they conjectured that
$$\begin{aligned} R(P_{n},P_{n},P_{n}) = \left\{ \begin{array}{ll} 2n-2 &{} \quad \text {if } n \text { is even},\\ 2n-1 &{} \quad \text {if } n \text { is odd}. \end{array} \right. \end{aligned}$$
In 2007 this conjecture was established by Gyárfás et al. [11] for sufficiently large n. We can apply our result for \(R(C_{n_0}, P_{n_{1}},P_{n_{2}})\) to \(P_{n_0}\) instead of \(C_{n_0}\) to obtain the same result as in [7]. The formula is the same but the original result contains the lower bound for \(n_0\) and no conditions for \(n_1\) and \(n_2\) (except \(n_1, n_2 \ge 2\)).

The Ramsey number of a star versus a path was completely determined by Parsons [15]. In Sects. 3.3 and 3.6 we investigate multicolor Ramsey number of a cycle \(C_n\) or path \(P_n\) versus stars and stripes for large value of n.

In [13] Maherani et al. proved that \(R(P_{3}, kK_{2}, tK_{2})=2k+t-1\) for \(k\ge t \ge 3\). In this paper we show that \(R(P_n,kK_{2},tK_{2})=n+k+t-2\) for large n. In addition we prove that for even k, \(R((k-1)K_{2},P_{k},P_{k})=3k-4\). For \(s< m-1<2s+1\) and \(t\ge m+s-1\), we obtain that \(R(tK_{2},P_{2s+1},P_{2m})=s+m+2t-2.\)

We also provide some new exact values or generalize known results for other multicolor Ramsey numbers of paths, cycles, stripes and stars versus other graphs.

2 Main Results

Theorem 2

For every graph H and bipartite graphs \(G_1, \ldots , G_k,\) we have
$$\begin{aligned} R(H,G_1,\ldots ,G_k)\le R(H, K_{b,b}), \end{aligned}$$
where \(b=b(G_1, \ldots , G_k)\).

Proof

Assume \(R(H, K_{b,b})=n\), we will show that for any coloring of the edges of the complete graph \(K_{n}\) by \(k+1\) colors there exists a color i for which the corresponding color class contains \(G_{i}\) as a subgraph.

Suppose that \(G=K_{n}\) is \((k+1)\)-edge colored such that G does not contain H of color 1. We show that there is a copy of \(G_{i}\) of color i in G for some \(2\le i\le k+1\). Now we merge k colors classes \(2, \ldots , k+1\). Suppose that new class is black. Since \(R(H, K_{b,b})=n\), we have a black copy of \(K_{b,b}\). Using to its original k-coloring, we see that there exists a complete bipartite subgraph \(L=K_{b,b}\) whose edges are colored with \(2,\ldots ,k+1\) (observe that there is no edges of color 1 in L). Now \(b=b(G_1, \ldots , G_k)\) guarantees that L contains a copy of \(G_{i}\) of color i for some \(2\le i\le k+1\).

Häggkvist [12] obtained the upper bound \(R(P_{m},K_{n,k})\le k+n+m-2.\) In addition, Faudree et al. [9] obtained the exact value \(R(tK_{2},K_{n,n})=max \lbrace n+2t-1,2n+t-1 \rbrace .\) Using these results with Theorem 2 we immediately obtain the following.

Corollary 1

For bipartite graphs \(G_1, \ldots , G_k,\) we have
  1. 1.
    $$\begin{aligned} R(P_{m}, G_1,\ldots ,G_k)\le 2b+m-2, \end{aligned}$$
     
  2. 2.
    $$\begin{aligned} R(tK_{2}, G_1,\ldots , G_k) \le {\left\{ \begin{array}{ll} 2b+t-1 &{} \text {if } t \le b, \\ b+2t-1 &{} \text {if } t \ge b, \end{array}\right. } \end{aligned}$$
    where \(b=b(G_1, \ldots , G_k)\).
     

For bipartite graphs \(G_1, G_2, \ldots , G_k\) we easily have \(R(G_1, G_2, \ldots , G_k) \le 2b(G_1, G_2, \ldots , G_k)\). The answer to the question when the equality holds is an open problem. For example, we know that \(b(C_{2m}, C_4) = m + 1\) for \( m \ge 4 \), while \(R(C_{2m}, C_4) = 2m + 1\) for \( m \ge 3 \) (see [18, 19], respectively).

Corollary 2

If \(R(G_1, G_2, \ldots , G_k) = 2b(G_1, G_2, \ldots , G_k)=2t\) for bipartite graphs \(G_1, \ldots , G_k,\) then
$$\begin{aligned} R(tK_{2},G_1,\ldots ,G_k)=3t-1. \end{aligned}$$

Proof

By Corollary 1, the upper bound is clear. To see the lower bound consider the graph \(G=K_{3t-2}=K_{2t-1}+ K_{t-1}\). Since \(R(G_1, \ldots , G_k)=2t\), we take a k-coloring of \(E(K_{2t-1})\) which does not contain a copy of \(G_{i}\) in color i for any \(1 \le i\le k\). The remaining edges of G we color with color \((k+1)\) and clearly such \(G^{k+1}\) contains no copy of \(tK_{2}\). This proves the corollary.

3 More Corollaries

This section contains a number of corollaries following from the main Theorem 2 and some known results about Ramsey numbers and bipartite Ramsey numbers.

3.1 \(R(C_{n_{0}},P_{n_{1}},P_{n_{2}})\) for Large \(n_{0}\)

In this subsection, we determine the value of \(R(C_{n_{0}},P_{n_{1}},P_{n_{2}})\) for large \(n_{0}\) and special cases of \(n_{1}, n_{2}\). We first recall a result of Bondy and Erdős from 1973 [1] for \(n> n_{1}(r)\) (that is for sufficiently large n). More precisely, they showed the following.

Theorem 3

[1] For \(n> n_{1}(r,t),\)\(R(C_{n},K^{t+1}_{r})=t(n-1)+r\) where \(K^{t+1}_{r}\) is the complete \((t+1)\)-partite graph with parts of size r. In particular \(R(C_{n},K_{r,r})=n+r-1\).

We can apply this result and Theorem 2 to determine the value which can be also obtained as consequence of Theorem 2.2 in [14].

Theorem 4

For sufficiently large \(n_{0},\) if \(n_{1}=2s\) and \(n_{2}=2m\) such that \(m-1<2s\) or if \(n_{1}=n_{2}=2s\) or if \(n_{1}=2s+1\) and \(n_{2}=2m\) such that \(s<m-1<2s+1,\) then we have
$$\begin{aligned} R(C_{n_0}, P_{n_{1}},P_{n_{2}}) = n_0 + \Big \lfloor \frac{n_1}{2} \Big \rfloor + \Big \lfloor \frac{n_2}{2} \Big \rfloor -2. \end{aligned}$$

Proof

For the upper bound, by Theorem 2, \(R(C_{n_{0}},P_{n_{1}},P_{n_{2}})\le R(C_{n_{0}},K_{b,b})\) where \(b=b(P_{n_{1}},P_{n_{2}})\). By Theorem 3, \(R(C_{n_{0}},P_{n_{1}},P_{n_{2}})\le n_{0}+b-1\). On the other hand, \(b=\Big \lfloor \frac{n_1}{2} \Big \rfloor + \Big \lfloor \frac{n_2}{2} \Big \rfloor -1\) (see Theorem 5) and \(R(C_{n_{0}},P_{n_{1}},P_{n_{2}}) \le n_{0}+\Big \lfloor \frac{n_1}{2} \Big \rfloor + \Big \lfloor \frac{n_2}{2} \Big \rfloor -2\).

For the lower bound, consider the graph \(G=K_{R-1}\cup K_{\frac{n_2}{2}-1}\) where \(R=R(C_{n_{0}},P_{n_{1}})\). It is known that \(R=n_{0}+\Big \lfloor \frac{n_1}{2} \Big \rfloor -1\) for \(n_{0}\ge n_{1}\ge 2\). It is clear that there is a blue/red coloring of \(K_{R-1}\) such that \(G^{b}\) contains no copy of \(C_{n_{0}}\) and \(G^{r}\) contains no copy of \(P_{n_{1}}\). Color the remaining subgraph \(K_{\frac{n_2}{2}-1}\) with red. Since \(\frac{n_2}{2}-1 < n_{1}\), there is no a red copy of \(P_{n_{1}}\) in \(K_{\frac{n_2}{2}-1}\). Consider \({\overline{G}}={\overline{K}}_{R-1}+ {\overline{K}}_{\frac{n_2}{2}-1}\) and color it with green. Thus \(G^{g}\) contains no copy of \(P_{n_{2}}\). The equality follows.

In 1975 Faudree and Schelp [7] proved that if \(n_{0}\ge 6(n_{1}+n_{2})^{2}\), then \(R(P_{n_{0}},P_{n_{1}},P_{n_{2}})=n_{0}+\lfloor n_{1}/2 \rfloor +\lfloor n_{2}/2 \rfloor -2\) for \(n_{1}, n_{2}\ge 2\). Since \(R(P_{n_{0}},P_{n_{1}},P_{n_{2}}) \le R(C_{n_{0}},P_{n_{1}},P_{n_{2}})\), we can apply Theorem 4 to \(P_{n_{0}}\) instead of \(C_{n_{0}}\) to obtain the same results as in [7].

Corollary 3

For sufficiently large \(n_{0},\) if \(n_{1}=2s\) and \(n_{2}=2m\) where \(m-1<2s\) or if \(n_{1}=n_{2}=2s\) or if \(n_{1}=2s+1\) and \(n_{2}=2m\) where \(s<m-1<2s+1,\) then we have
$$\begin{aligned} R(P_{n_0}, P_{n_{1}},P_{n_{2}}) = n_0 + \Big \lfloor \frac{n_1}{2} \Big \rfloor + \Big \lfloor \frac{n_2}{2} \Big \rfloor -2. \end{aligned}$$

Proof

Since \(R(P_{n_0}, P_{n_1}) = n_{0}+\Big \lfloor \frac{n_1}{2} \Big \rfloor -1\) for \(n_{0}\ge n_{1}\ge 2\), let us consider the same graph and coloring as in the proof of Theorem 4.

3.2 \(R(tK_{2},P_{k},P_{k^{'}})\)

In 1975 Faudree and Schelp [8] determined \(b(P_{n},P_{k})\) for all n and k. In the following theorem we present only two cases which we will use in the proof of the next theorem.

Theorem 5

[8] For sm positive integers, 
  1. 1.

    \(b(P_{2s},P_{2m})=s+m-1,\)

     
  2. 2.

    \(b(P_{2s+1},P_{2m})= s+m-1\) for \(s < m-1\).

     

Theorem 6

For positive integers kmst
  1. (i)

    \(R((k-1)K_{2},P_{k},P_{k})=3k-4\) if k is even, 

     
  2. (ii)

    \(R(tK_{2},P_{2s+1},P_{2m})=s+m+2t-2\) for \(s< m-1<2s+1\) and \(t\ge m+s-1\).

     

Proof

(i) Corollary 1 implies that \(R(tK_{2},P_{k},P_{k}) \le 2b+t-1\) for \(t\le b\) and \(b=b(P_{k},P_{k})\). Theorem 5 (that is \(b=b(P_{k},P_{k})=k-1\) for even k) completes the proof for the upper bound.

It is known that \(R=R((k-1)K_{2},P_{k})=2k+\lfloor k/2 \rfloor -3\) (see [9]). Consider the graph \(G=K_{R-1}\cup K_{k/2 -1}\). Clearly \(K_{R-1}\) has a blue/red coloring such that \(K_{R-1}^{b}\) contains no copy of \((k-1)K_{2}\) and \(K_{R-1}^{r}\) contains no copy of \(P_{k}\). We can take this coloring and we color the edges of \(K_{k/2 -1}\) with red. The edges of \({\overline{G}}\) we color with green. Hence \(G^{g}\) contains no copy of \(P_{k}\). This gives the desired lower bound.

(ii) As before, by Corollary 1, we have \(R(tK_{2},P_{2s+1},P_{2m}) \le b+2t-1\) for \(t\ge m+s-1\). On the other hand by Theorem 5, for \(s < m-1\), \(b(P_{2s+1},P_{2m})=s+m-1\). So for \(s < m-1\) and \(t\ge m+s-1\), \(R(tK_{2},P_{2s+1},P_{2m}) \le s+m+2t-2\).

Now, let \(G=K_{R-1}\cup K_{m-1}\) where \(R=R(tK_{2}, P_{2s+1})=2t+\lfloor (2s+1)/2 \rfloor -1\) for \(t> \lfloor (2s+1)/2 \rfloor \) (see [9]). Clearly \(K_{R-1}\) can be colored in such a way that \(K_{R-1}^{b}\) contains no copy of \(tK_{2}\) and \(K_{R-1}^{r}\) contains no copy of \(P_{2s+1}\). We color the subgraph \(K_{m-1}\) with red and the edges of \({\overline{G}}\) with green. Then \(G^{g}\) contains no copy of \(P_{2m}\) and the proof is complete.

3.3 \(R(C_n,kK_{2},tK_{2})\) and \(R(P_n,kK_{2},tK_{2})\) for Large n

Lemma 1

[3] For positive integers m and n
$$\begin{aligned} b(mK_{2}, nK_{2})=m+n-1. \end{aligned}$$

Theorem 7

\(R(C_n,kK_{2},tK_{2})= n+k+t-2,\) for sufficiently large n.

Proof

Theorems 23 and Lemma 1 give us the desired upper bound.

In [9] it is shown that \(R(C_{n}, kK_{2})=n+k-1\) for \(k \le \lfloor \frac{n}{2} \rfloor \). Consider the graph \(G=K_{n+k-2} \cup {\overline{K}}_{t-1}\). There is a blue/red coloring of \(K_{n+k-2}\) such that \(G^{b}\) contains no copy of \(C_{n}\) and \(G^{r}\) contains no copy of \(kK_{2}\). Color \({\overline{G}}={\overline{K}}_{n+k-2}+K_{t-1}\) with color green. The theorem follows.

In [13] Maherani et al. proved that \(R(P_{3}, kK_{2}, tK_{2})=2k+t-1\) for \(k\ge t\ge 3\).

Theorem 8

\(R(P_n,kK_{2},tK_{2})= n+k+t-2,\) for sufficiently large n.

Proof

Clearly \(R(P_n,kK_{2},tK_{2})\le R(C_{n},kK_{2},tK_{2})\). Theorem 7 gives us the upper bound.

In [9] it is shown that \(R(P_{n}, kK_{2})=n+k-1\) for \(k \le \lfloor \frac{n}{2} \rfloor \). We obtain the lower bound by considering the same coloring as in the proof of Theorem 7.

3.4 \(R(tK_{2},P_{3},C_{2n})\) for \(t \le n\)

Theorem 9

[18] \(R(P_{3},C_{2n})=2n\).

Theorem 10

For positive integers \(t \le n,\)\(R(tK_{2},P_{3},C_{2n})= 2n+t-1\).

Proof

It is easy to see that \(b(P_{3},C_{2n})=n\) for \(n\ge 3\). By Corollary 1, we have \(R(tK_{2},C_{2n},P_{3})\le 2n+t-1\) where \(t \le n\). To see the lower bound, assume that \(G=K_{2n-1}\cup {\overline{K}}_{t-1}\). It is clear that there is a blue/red coloring of \(E(K_{2n-1})\) such that there is no blue copy \(C_{2n}\) and no red copy \(P_{3}\). We can take this coloring and we color the edges of \({\overline{G}}={\overline{K}}_{2n-1}+ K_{t-1}\) with color green. There is no green copy \(tK_{2}\).

3.5 \(R(tK_{2},C_{2m},C_{4})\)

Theorem 11

[19] \(b(C_{2m},C_{4})=m+1\) for \(m\ge 4.\)

Theorem 12

[9] \(R(tK_{2},C_{n})=max \lbrace n+2t-1- \lfloor n/2 \rfloor , n+t-1 \rbrace \) for \(n\ge 3.\)

Theorem 13

For a positive integer \(m\ge 4,\)
  1. 1.

    if \(t \ge m+1\) then \(R(tK_{2},C_{2m},C_{4})=m+2t,\)

     
  2. 2.

    if \(t \le m\) then \(2m+t \le R(tK_{2},C_{2m},C_{4})\le 2m+t+1\).

     

Proof

(1) By Corollary 1 and Theorem 11, we have \(R(tK_{2},C_{2m},C_{4})\le b+2t-1=m+2t\) where \(b=b(C_{2m},K_{2,2})\) and \(t \ge m+1\). To see the lower bound, consider the graph \(G=K_{R-1}\cup K_{1}\) where \(R=R(tK_{2},C_{2m})=m+2t-1\) for \(t \ge m+1\). It is clear that there is a blue/red coloring of \(E(K_{R-1})\) such that there is no blue copy of \(tK_{2}\) and no red copy of \(C_{2m}\). We can take this coloring. We color the edges of \({\overline{G}}={\overline{K}}_{R-1}+ {\overline{K}}_{1}\) by green. So there is no green copy of \(C_{4}\).

(2) By Corollary 1, we have \(R(tK_{2},C_{2m},C_{4})\le 2b+t-1\) where \(b=b(C_{2m},K_{2,2})\) and \(t\le m+1\). By Theorem 11, we obtain \(R(tK_{2},C_{2m},C_{4})\le 2m+t+1\). To see the lower bound, consider the graph \(G=K_{R-1}\cup K_{1}\) where \(R=R(tK_2,C_{2m})=2m+t-1\). It is clear that there is a blue/red coloring of \(E(K_{R-1})\) such that there is no blue copy of \(tK_{2}\) and no red copy \(C_{2m}\). We can take such a coloring and we color the edges of \({\overline{G}}={\overline{K}}_{R-1}+ {\overline{K}}_{1}\) with green. There is no green copy of \(C_4\) and the proof is complete.

3.6 Multicolor Ramsey Numbers

This last subsection contains some results for Ramsey numbers with more than 3 colors.

Theorem 14

Let \(m_{1}, m_{2},\ldots ,m_{s}\) and \(k_{1}, k_{2}, \ldots , k_{t}\) be positive integers and \(n> n_{1}(b)\) where \(b=b(K_{1,k_{1}}, K_{1,k_{2}}, \ldots , K_{1,k_{t}}, m_{1}K_{2}, m_{2}K_{2}, \ldots , m_{s}K_{2})\). Then
$$\begin{aligned} R(C_n,K_{1,k_{1}},K_{1,k_{2}}, \ldots ,K_{1,k_{t}}, m_{1}K_{2}, m_{2}K_{2}, \ldots , m_{s}K_{2})\le n+b-1. \end{aligned}$$

Proof

By using the same argument as in Theorem 2, we have \(R(C_n,K_{1,k_{1}}, K_{1,k_{2}},\)\(\ldots , K_{1,k_{t}}, m_{1}K_{2}, m_{2}K_{2}, \ldots , m_{s}K_{2}) \le R(C_n,K_{b,b})\) where \(b=b(K_{1,k_{1}}, K_{1,k_{2}},\)\(\ldots , K_{1,k_{t}}, m_{1}K_{2}, m_{2}K_{2}, \ldots , m_{s}K_{2})\). Next we apply Theorem 3.

Lemma 2

[16] Let \(m_{1}, m_{2},\ldots ,m_{s}\) and \(k_{1}, k_{2}, \ldots , k_{t}\) be positive integers with \(\varLambda =\sum _{i=1}^{s}(m_{i}-1)\) and \(\sum =\sum _{i=1}^{t}(k_{i}-1)\). Then \(b(K_{1,k_{1}}, K_{1,k_{2}}, \ldots , K_{1,k_{t}}, m_{1}K_{2}, m_{2}K_{2},\)\(\ldots , m_{s}K_{2})=b,\) where
$$\begin{aligned} b = {\left\{ \begin{array}{ll} \varLambda +1 &{} \text {if } \varSigma <\lfloor (\varLambda +1)/2 \rfloor , \\ \varSigma +\lfloor \varLambda /2 \rfloor +1 &{} \text {if } \varSigma \ge \lfloor (\varLambda +1)/2 \rfloor . \end{array}\right. } \end{aligned}$$

Combining Theorem 14 and Lemma 2, we obtain the following.

Theorem 15

Let \(m_{1}, m_{2},\ldots ,m_{s}\) and \(k_{1}, k_{2}, \ldots , k_{t}\) be positive integers with \(\varLambda =\sum _{i=1}^{s}(m_{i}-1)\) and \(\sum =\sum _{i=1}^{t}(k_{i}-1)\). If \(\varSigma \le \lfloor (\varLambda +1)/2 \rfloor \) for \(n> n_{1}(\varLambda ),\) then
$$\begin{aligned} R(C_n,m_{1}K_{2},m_{2}K_{2},\ldots ,m_{s}K_{2},K_{1,k_{1}},K_{1,k_{2}},\ldots ,K_{1,k_{t}})= n+\varLambda . \end{aligned}$$

Proof

For the upper bound, we apply Theorem 14 and Lemma 2.

For the lower bound color all edges of \(G=K_{n-1}\cup {\overline{K}}_{\varLambda }\) by color 1 and for all edges of \({\overline{G}}={\overline{K}}_{n-1}+ K_{\varLambda }\) consider the following coloring. Color \({\overline{K}}_{n-1}+ K_{m_{1}-1}\) and \({\overline{K}}_{m_{1}-1}+{\overline{K}}_{\sum _{i=2}^{s}(m_{i}-1)}\) by color 2 and color \({\overline{K}}_{n-1}+ K_{m_{2}-1}\) and \({\overline{K}}_{m_{2}-1}+{\overline{K}}_{\sum _{i=3}^{s}(m_{i}-1)}\) by color 3 and in the general color \({\overline{K}}_{n-1}+ K_{m_{j}-1}\) and \({\overline{K}}_{m_{j}-1}+{\overline{K}}_{\sum _{i=j+1}^{s}(m_{i}-1)}\) by color \(j+1\) where \( j\ge 1\) and finally color \({\overline{K}}_{n-1}+ K_{m_{s}-1}\) by color s. Then \(G^{1}\) contains no copy of \(C_{n}\), \(G^{i+1}\) contains no copy of \(m_{i}K_{2}\) for \(1\le i \le s\), as desired. Note that this lower bound is usable on arbitrary graphs \( H_{1},\ldots , H_{t}\) instead of \(K_{1,k_{1}},K_{1,k_{2}},\ldots ,K_{1,k_{t}}\).

Corollary 4

For given positive integers \(m_{1}, m_{2},\ldots ,m_{s}\) and \(k_{1},k_{2},\ldots ,k_{t}\) with \(\varLambda =\sum _{i=1}^{s}(m_{i}-1),\)\(\sum =\sum _{i=1}^{t}(k_{i}-1),\) sufficiently large n and \(\varSigma \le \lfloor (\varLambda +1)/2 \rfloor ,\)
$$\begin{aligned} R(P_{n},K_{1,k_{1}},K_{1,k_{2}},\ldots ,K_{1,k_{t}},m_{1}K_{2},m_{2}K_{2},\ldots ,m_{s}K_{2}) =n+\varLambda . \end{aligned}$$

Lemma 3

[16] For positive integers \(m, k_{1},\ldots ,k_{r}\ge 2\) and \(\sum =\sum _{i=1}^{r}(k_{i}-1),\)
$$\begin{aligned} b(P_{m},K_{1,k_{1}},\ldots ,K_{1,k_{r}})= {\left\{ \begin{array}{ll} \varSigma + \frac{m}{2} &{} \text {if } \varSigma \ge \frac{m}{2}, m\text { even,}\\ \varSigma + \frac{m+1}{2}&{} \text {if } \varSigma \ge \frac{m-1}{2}, m\text { odd, } \varSigma \equiv 0 \text { mod}(\frac{m-1}{2}),\\ \varSigma + \frac{m-1}{2} &{} \text {if } \varSigma \ge \frac{m-1}{2}, m\text { odd, } \varSigma \not \equiv 0 \text { mod}(\frac{m-1}{2}),\\ 2\varSigma +1 &{} \text {if } \frac{1}{2}\lfloor \frac{m}{2} \rfloor +1 \le \varSigma< \lfloor \frac{m}{2} \rfloor +1,\\ \lfloor \frac{m+1}{2} \rfloor &{} \text {if } \varSigma < \frac{1}{2}\lfloor m/2 \rfloor . \end{array}\right. } \end{aligned}$$

Theorem 16

For \( \varSigma <\frac{s}{2},\)\(m=2s\) and \(2 \le t\le s,\)
$$\begin{aligned} R(tK_{2},P_{m},K_{1,k_{1}},\ldots ,K_{1,k_{r}})=m+t-1. \end{aligned}$$

Proof

Suppose that \( \varSigma < \frac{s}{2}\), \(m=2s\) and \(2 \le t\le s\). By Corollary 1 and Lemma 3, we have \(R(tK_{2},P_{m},K_{1,k_{1}},\ldots ,K_{1,k_{r}})\le m+t-1 \). To obtain the lower bound, divide the vertex set of \(G=K_{m+t-2}\) into two parts A and B, where \(|A|=m-1\) and \(|B|=t-1\). Color the edges of G[A] with the second color and other edges with the first color. Note that the lower bound is usable for arbitrary graphs \( H_{1},\ldots , H_{t}\) instead of \(K_{1,k_{1}},K_{1,k_{2}},\ldots ,K_{1,k_{r}}\).

Notes

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
  2. 2.Institute of Informatics, Faculty of Mathematics, Physics and InformaticsUniversity of GdańskGdańskPoland

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