On size multipartite Ramsey numbers for stars

Burger and Vuuren deﬁned the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs m j ( K a × b , K c × d ) , for natural numbers a, b, c, d and j, where a, c ≥ 2 , in 2004. They have also determined the necessary and sufﬁcient conditions for the existence of size multipartite Ramsey numbers m j ( K a × b , K c × d ) . Syafrizal et. al. generalized this deﬁnition by removing the completeness requirement. For simple graphs G and H, they deﬁned the size multipartite Ramsey number m j ( G, H ) as the smallest natural number t such that any red-blue coloring on the edges of K j × t contains a red G or a blue H as a subgraph. In this paper, we determine the necessary and sufﬁcient conditions for the existence of multipartite Ramsey numbers m j ( G, H ) , where both G and H are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers m j ( K 1 ,m , K 1 ,n ) for all integers m, n ≥ 1 and j = 2 , 3 , where K 1 ,m is a star of order m + 1 . In addition, we also determine the lower bound of m 3 ( kK 1 ,m , C 3 ) , where kK 1 ,m is a disjoint union of k copies of a star K 1 ,m and C 3 is a cycle of order 3.


Introduction
The classical Ramsey number r(a, c) is the smallest natural number j such that any red-blue coloring of the edges of K j , necessarily forces a red K a or a blue K c as subgraph. The size multipartite Ramsey number is one of generalizations of the classical Ramsey number. Burger and Vuuren [1] gave a definition of the size multipartite Ramsey numbers for a pair of complete, balanced, multipartite graphs, as follows. Let a, b, c, d and j, be natural numbers with a, c ≥ 2, the size multipartite Ramsey number m j (K a×b , K c×d ) is the smallest natural number t such that any red-blue coloring of the edges of K j×t , necessarily forces a red K a×b or a blue K c×d as subgraph. They also determined m j (K 2×2 , K 3×1 ), for j ≥ 1 and have established the following existence of size multipartite Ramsey numbers. Theorem 1.1. (The existence of size numbers) [1] The size multipartite Ramsey numbers m j (K a×b , K c×d ) exists for any a, c ≥ 2 and b, d ≥ 1 if and only if j ≥ r(a, c).
Syafrizal et. al. [10] generalized this definition by removing the completeness requirement. For simple graphs G and H, they defined the size multipartite Ramsey number m j (G, H) as the smallest natural number t such that any red-blue coloring on the edges of K j×t contains a red G or a blue H as a subgraph. The size bipartite Ramsey numbers for stars versus paths m 2 (K 1,m , P n ), for m, n ≥ 2 given by Hattingh and Henning [3]. In 2007, Syafrizal et al. [11] determined the size multipartite Ramsey numbers for stars versus P 3 . Then, Surahmat et al. [9] gave the size tripartite Ramsey numbers for stars versus P n , for 3 ≤ n ≤ 6. Furthermore, we gave the size multipartite Ramsey numbers for stars versus cycles [5] and the size tripartite Ramsey numbers for a disjoint union of m copies of a star K 1,n versus P 3 [6]. In 2017, Jayawardene et al. [4] and Effendi et al. [2] determined the size multipartite Ramsey numbers for stars versus paths. Then, we also gave the size multipartite Ramsey numbers for stars versus paths and cycles [7], that complete the previous results given by Syafrizal and Surahmat. Recently, we determined m j (mK 1,n , H), where H = P 3 or K 1,3 for j ≥ 3, m, n ≥ 2 [8].
In this paper, we determine the necessary and sufficient conditions for the existence of the size multipartite Ramsey numbers m j (G, H), where both G and H are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers m j (K 1,m , K 1,n ) for all integers m, n ≥ 1 and j = 2, 3. In addition, we also determine the lower bound of m 3 (kK 1,m , C 3 ).
We call some basic definitions that will be used in this paper, as follows. Let G be a finite and simple graph. Let vertex and edge sets of graph G are denoted by V (G) and E(G), respectively. Vertex colorings in which adjacent vertices are colored differently are proper vertex colorings. A graph G is k-colorable if there exists a proper vertex coloring of G from a set of k colors. A matching of a graph G is defined as a set of edges without a common vertex. A matching of maximum size in G is a maximum matching in G. The maximum degree of G is denoted by n is the graph on n + 1 vertices with one vertex of degree n, called the center of this star, and n vertices of degree 1, called the leaves. A disjoint union of k copies of a star K 1,m , a cycle of order n, and a path of order n are denoted by kK 1,m , C n , and P n , respectively.

Results
For any non complete graphs G and H, we will determine the necessary and sufficient conditions for the existence of the size multipartite Ramsey numbers m j (G, H). In order to do so, we recall the definition of the chromatic number of a graph G, denoted by χ(G), which is the minimum positive integer k for which G is k-colorable.
Lemma 2.1. In every proper vertex coloring of a simple graph G, the maximum number of the vertices in G with the same color is |V (G)| − χ(G) + 1.
Proof. Let c be a proper vertex coloring of G, with χ(G) color, that is c :  Proof. We will show that m 2 (K 1,m , K 1,n ) ≥ m + n − 1. We consider a red-blue coloring on the edges of graph K 2×(m+n−2) = F R ⊕F B , such that F R is a (m−1)−regular graph. By Handshaking Lemma, it is possible since the sum of the degrees of the vertices of F R is even. Then, F R K 1,m . We have d(v) = m + n − 2 − (m − 1) = n − 1, for any v in F B . Hence, F B K 1,n . Now, we will show that m 2 (K 1,m , K 1,n ) ≤ m+n−1. We consider any red-blue coloring on the edges of graph K 2×(m+n−1) = G R ⊕G B , such that G R K 1,m . This implies that ∆(G R ) ≤ m−1. Therefore, δ(G B ) ≥ m + n − 1 − (m − 1) = n. Then, G B ⊇ K 1,n . Theorem 2.3. For positive integers m and n, we have for m ≡ 2 mod 4, n = 1, 2 2 m+1 For n = 1, we will use the property that m 3 (K 1,m , K 1 ) ≤ m 3 (K 1,m , K 1,1 ). It is clear that For m ≡ 6 mod 4 and n = 2, we consider a red-blue coloring on the edges of graph K 3×( m 2 −1) , such that K 3×( m 2 −1) contains a maximum blue matching graph. Since m 2 − 1 is even, the blue graph is a 1−regular graph. This implies that graph K 3×( m 2 −1) contains red (m − 3)−regular graph. So K 3×( m 2 −1) contains no a red K 1,m . Then, m 3 (K 1,m , K 1,2 ) ≥ m 2 . Furthermore, we consider any red-blue coloring on the edges of graph K 3× m 2 , such that graph K 3× m 2 contains no a blue K 1,2 . This implies that the maximum degree of blue graph is 1.
By Handshaking Lemma, it is possible since the sum of the degrees of the vertices of F R is even. Then, F R K 1,m . We distinguish the following two cases, to show that m 3 (K 1,m , K 1,n ) ≥ t.
Case a. For m ≡ 2 mod 4 and n ≡ 3 mod 4. Now, we consider any red-blue coloring on the edges of graph K 3×t = G R ⊕ G B , such that G R K 1,m . This implies that ∆(G R ) ≤ m − 1. We distinguish the following two cases, to show that m 3 (K 1,m , K 1,n ) ≤ t.