Connected size Ramsey number for matchings vs. small stars or cycles

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1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 127, No. 5, November 2017, pp Connected size Ramsey number for matchings vs. small stars or cycles BUDI RAHADJENG, EDY TRI BASKORO and HILDA ASSIYATUN Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung (ITB), Bandung, Indonesia MS received 1 September 2016; revised 18 September 2017; published online 20 November 2017 Abstract. The notation F (G, H) means that if the edges of F are colored red and blue, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The connected size Ramsey number ˆr c (G, H) of graphs G and H is the minimum size of a connected graph F satisfying F (G, H). Form 2, the graph consisting of m independent edges is called a matching and is denoted by mk 2. In 1981, Erdös and Faudree determined the size Ramsey numbers for the pair (mk 2, K 1,t ).Theyshowed that the disconnected graph mk 1,t (mk 2, K 1,t ) for t, m 1. In this paper, we will determine the connected size Ramsey number ˆr c (nk 2, K 1,3 ) for n 2andˆr c (3K 2, C 4 ). We also derive an upper bound of the connected size Ramsey number ˆr c (nk 2, C 4 ), for n 4. Keywords. Connected size Ramsey number; cycle; matching; star Mathematics Subject Classification. 05D10, 05C Introduction All graphs in this paper are finite, simple and undirected. Let F, G and H be graphs. The notation F (G, H) means that in any red blue coloring of the edges of F there exists a red copy of G or a blue copy of H in F. We denote F (G, H) to mean that there is some red blue coloring of the edges of F such that F contains neither a red G nor a blue H. This coloring is defined as the (G, H)-coloring of F. The size Ramsey number of a pair graph G and H, denoted by ˆr(G, H), is the least integer k such that there is a graph F with k edges satisfying F (G, H). The size Ramsey number of a graph was introduced by Erdös et al. in [2]. A survey of results concerning the size Ramsey number for many pairs of graphs can be seen in [3]. For m 2, the graph consisting of m independent edges is called a matching and is denoted by mk 2. In [1], Erdös and Faudree have determined the size Ramsey numbers for a pair of graphs where one of which is a matching. Some of the results obtained are as follows. Theorem 1.1 [1].Fort, n 1, ˆr(tK 2, K 1,n ) = tn. Indian Academy of Sciences 787

2 788 Budi Rahadjeng et al. Theorem 1.2 [1]. For fixed t 2, there are positive constants a and b such that for all n 3, n + a n < ˆr(tK 2, C n )<n + b n. The graph achieving the size Ramsey number ˆr(tK 2, K 1,n ) is tk 1,n and the graph achieving the size Ramsey number ˆr(2K 2, C 4 ) is 2C 4. Both graphs are disconnected. Motivated by the above resuls, in this paper, we focus on finding a connected graph F with minimum size satisfying F (G, H). In particular, we consider G = tk 2 and H is a star or a cycle. We define the connected size Ramsey number, denoted by ˆr c (G, H) and defined as follows: ˆr c (G, H) = min{ E(F) :F (G, H), F is connected}. Rahadjeng et al. [5] have established an upper bound of the connected size Ramsey numbers for nk 2 and K 1,m as follows. Lemma 1.3 [5]. For positive integers n 2, m 3, ˆr c (nk 2, K 1,m ) nm + (n 1). In [4], Rahadjeng et al. showed the following theorem. Theorem 1.4 [4].Forn 4, ˆr c (2K 2, C n ) = 2n. Continuing the above results, in this paper, we will determine the exact value of the connected size Ramsey number ˆr c (nk 2, K 1,3 ) for n 2 and ˆr c (3K 2, C 4 ). We also derive an upper bound of the connected size Ramsey number for nk 2 versus C 4, for n Main results The main results are presented in the following theorems. Lemma 2.1. ˆr c (2K 2, K 1,3 ) = 7. Proof. By Lemma 1.3, we obtain that ˆr c (2K 2, K 1,3 ) 7. Now, we will show that ˆr c (2K 2, K 1,3 ) 7. Let F be a connected graph with E(F) 6. Suppose that (F) 4. Let v V (F) with d(v) 4 and set F = F v. Observe that E(F ) 2. Then the edges of F are colored blue and all edges incident with v are colored red. By this coloring, there exists a (2K 2, K 1,3 )-coloring of F. So, F (2K 2, K 1,3 ). Now, assume that (F) = 3. Let v V (F) with d(v) = 3 and set F = F v. Observe that E(F ) 3. If F contains no K 1,3 then all edges in F are colored blue. By coloring red all edges incident to v, we obtain a (2K 2, K 1,3 )-coloring of F. So, F (2K 2, K 1,3 ). Suppose F contains a K 1,3. Since F is a connected graph, then there exists a vertex x of F adjacent to v. Then, color all edges incident to x by red and the other edges of F by blue. This coloring is a (2K 2, K 1,3 )-coloring of F. So, F (2K 2, K 1,3 ). Now, if (F) = 2 then F must be a path or a cycle. Clearly, by coloring all edges of F by blue, we have that a (2K 2, K 1,3 )-coloring of F F (2K 2, K 1,3 ).

3 Connected size Ramsey number 789 Lemma 2.2. ˆr c (3K 2, K 1,3 ) = 11. Proof. By Lemma 1.3, we obtain ˆr c (3K 2, K 1,3 ) 11. Now, we will show that ˆr c (3K 2, K 1,3 ) 11. Let F be a connected graph with E(F) 10. Suppose that (F) 4. Let v V (F) with d(v) 4 and F = F v. Observe that E(F ) 6. If F is a connected graph, then by Lemma 2.1, F (2K 2, K 1,3 ).Now, color all edges in F incident with v by red. Therefore, we obtain a (3K 2, K 1,3 )-coloring on F. So, F (3K 2, K 1,3 ). Let F be a disconnected graph. If all the components contain no K 1,3, then color all edges of F by blue and other edges of F by red, we obtain F (3K 2, K 1,3 ). Now, we assume that at least one of the components contains K 1,3. Since E(F ) 6, there are at most two components containing K 1,3 in F. If only one component F 1 of F contain K 1,3, then F 1 has at most 6 edges. In case the number of edges of F 1 is exactly 6, the other component is an isolated vertex. By Lemma 2.1, there is a (2K 2, K 1,3 )-coloring on F 1. Therefore, by coloring all edges incident with v by red and all edges in the other component by blue, we obtain a (3K 2, K 1,3 )-coloring on F. So, F (3K 2, K 1,3 ). Now assume that the two components of F contain K 1,3. In this case, since F is connected, there is at least one vertex in each component of F adjacent to v. Now, color all edges incident to these two vertices by red and the remaining edges of F by blue. Then, we obtain a (3K 2, K 1,3 )-coloring on F. So, F (3K 2, K 1,3 ). Now, assume that (F) = 3. Since F is a connected graph with E(F) 10, by handshaking lemma, the number of vertices of degree 3 in F is at most 6. If there are at most two vertices of degree 3 in F, then color all edges incident with these two vertices by red and the remaining edges by blue. We obtain F (3K 2, K 1,3 ). Now, suppose there are at least three vertices of degree 3. Let A be the set of vertices of degree 3 in F. Since F is connected, there is a pair of vertices x and y in A, such that N(x) N(y) =.Letn 0 N(x) N(y). Color all edges incident with n 0 by red. Next, we consider the subgraph F = F n 0.IfF contains only one vertex of degree 3, then all edges incident with the vertex of degree 3 are colored by red and the other edges are colored with blue. Otherwise, a path joining the vertices of degree 3 is colored with red. Of course, the length of this path is at most 2. By this coloring, we obtain a (3K 2, K 1,3 )-coloring of F. Thus, F (3K 2, K 1,3 ). Next, suppose that (F) 2. Color all edges of F by blue. Then we have (3K 2, K 1,3 )- coloring of F. So, F (3K 2, K 1,3 ). Hence, in all cases, we obtain F (3K 2, K 1,3 ) and this concludes the proof. Theorem 2.3. For n 4, ˆr c (nk 2, K 1,3 ) = 4n 1. Proof. By Lemma 1.3, we obtain ˆr c (nk 2, K 1,3 ) 4n 1. Now, we will show that ˆr c (nk 2, K 1,3 ) 4n 1. Let F be a connected graph with E(F) 4n 2. We will show that F (nk 2, K 1,3 ). We will split the proof into two cases. Case 1. (F) 2. Color all edges of F by blue. Then we have (nk 2, K 1,3 )-coloring of F. So, F (nk 2, K 1,3 ). Case 2. (F) 3. Let M be the set of vertices of degree at least 3 in F. If M n 1, then color all edges incident with all vertices in M by red and the other edges by blue. By this coloring, there is a (nk 2, K 1,3 )-coloring in F.

4 790 Budi Rahadjeng et al. Figure 1. Graph F. Now, we assume that M n. Since E(F) 4n 2, there are at most (n 1) disjoint K 1,m, m 3inF, otherwise F has at least 4n 1 edges, a contradiction. Suppose that G contains at most (n 1) disjoint K 1,m, m 3. Let N be the set of the center vertex of disjoint K 1,m, m 3. Observe that the remaining edges of F are at least 3. Let us consider the remaining edges. If these edges induce no K 1,3, then we color all edges incident with all vertices of N with red and the other edges with blue. Now, suppose that these edges induce K 1,3 with center x i. Since F is connected, then at least one vertex of this K 1,3 adjacent to vertices in N. Therefore, x i and the vertices in N having distance at most 2. If x i is adjacent to a vertex of N, then color all edges incident with x i by red. Next, we color all edges incident with the remaining vertex of N with red and the remaining edges with blue. By this coloring, there is a (nk 2, K 1,3 )-coloring in F. Suppose x i is not adjacent to a vertex of N. Choose a path P 3 having the internal vertex of the largest degree connecting x i and a vertex of N. Color all edges incident to these internal vertices with red. Next, we color all edges incident with the remaining vertex in N with red and the remaining edges in F with blue. By this coloring, there is a (nk 2, K 1,3 )-coloring in F. Hence, in all cases, we obtain F (nk 2, K 1,3 ) and this completes the proof of the theorem. Lemma 2.4. ˆr c (3K 2, C 4 ) = 13. Proof. First, we will show that ˆr c (3K 2, C 4 ) 13. Consider the graph F in figure 1. The number of edges of F is 13. Let μ be any 2-coloring of F by red and blue such that there is no red 3K 2. This coloring implies that the connected subgraph induced by red edges in F forms a path with length i, 1 i 4 or a graph G 1 G 2 where G i {P 2, P 3, K 1,3, K 1,4 }. Observe that, for every pair vertices in F, sayx and y, F {x, y} always contains C 4. Similarly, F\P i+1, 1 i 4 contains C 4. As a consequence, we always obtain a blue C 4 in F. So, F (3K 2, C 4 ). Next, we will prove that ˆr c (3K 2, C 4 ) 13. To do this, we will show that for any connected graph G, with E(G) 12, there is a (3K 2, C 4 )-coloring in G. Suppose G be a connected graph with E(G) 12. If G contains no C 4, then color all edges of G by blue. By this coloring, we obtain (3K 2, C 4 )-coloring in graph G.IfG contains exactly

5 Connected size Ramsey number 791 Figure 2. Graph F. one C 4, then choose one vertex in C 4, say v, and color all edges incident with v by red. By this coloring, we certainly got (3K 2, C 4 )-coloring in graph G. Now, if G contains more than one C 4 then there exists at most two disjoint 4-cycle in G. If there is a common vertex contained in all cycles, then color all edges incident with those vertices by red and the remaining edges by blue. By this coloring, we will obtain (3K 2, C 4 )-coloring in graph G. Otherwise, let C4 1 and C2 4 are two disjoint 4-cycle in G. Since E(G) 12, there is at least one edges of E(G)\{E(C4 1) E(C2 4 )} which is connecting a vertex of C4 1 and C2 4. While the remaining edges, its end vertices can incident with a vertex of C4 1 or a vertex of C2 4 or both. Select two vertices x C1 4 and y C4 2 such that they are contained in most numbers of 4-cycles. Then, color all edges incident with x and y by red and the other edges in G by blue. By this coloring, we will obtain (3K 2, C 4 ) coloring in graph G. So, in all cases G (3K 2, C 4 ). As a consequence ˆr c (3K 2, C 4 ) 13. Lemma 2.5. For n 4, ˆr c (nk 2, C 4 ) 5n 1. Proof. To show that ˆr c (nk 2, C 4 ) 5n 1, we show that there exists a connected graph F so that F (nk 2, C 4 ). Consider the graph F depicted in figure 2. The graph F is formed from a path with n vertices and n disjoint 4-cycles by identifying each vertex of the path with a fixed vertex of a 4-cycle The number of edges of F is E(F) = 5n 1. Let μ be any 2-coloring of edges of F such that there is no blue C 4 in F. Since F contains no blue C 4, then in each C 4 in F, there exist at least one red edge. Thus, in total we have at least n red edges in F. Hence, F contains a red nk 2 and so F (nk 2, C 4 ). Acknowledgements This research was supported by Research Grant Program Riset dan Inovasi KK-ITB, Ministry of Research, Technology and Higher Education, Indonesia. References [1] Erdös P and Faudree R J, Size Ramsey numbers involving matching, Colloquia Math. Societatis János Bolyai 37 (1981) [2] Erdös P, Faudree R J, Rousseau C C and Schelp R H, The size Ramsey number, Period. Math. Hungar. 9 (1978) [3] Faudree R J and Schelp R H, A survey of results on the size Ramsey number, Paul Erdös Math. II 10 (2002)

6 792 Budi Rahadjeng et al. [4] Rahadjeng B, Baskoro E T and Assiyatun H, Connected size Ramsey numbers for matchings versus cycles or path, Proc. Comput. Sci. 74 (2015) [5] Rahadjeng B, Baskoro E T and Assiyatun H, Connected size Ramsey numbers of matchings and stars, AIP Conf. Proc (2016) , Communicating Editor: Sharad S Sane