On the Union of Graphs Ramsey Numbers

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1 Applied Mathematical Sciences, Vol. 8, 2014, no. 16, HIKARI Ltd, On the Union of Graphs Ramsey Numbers I Wayan Sudarsana Combinatorial and Applied Mathematics Research Group (CAMRG) Tadulako University Jalan Soekarno-Hatta Km. 9 Palu 94118, Indonesia Edy Tri Baskoro Combinatorial Mathematics Research Group Bandung Institute of Technology Jalan Ganesha No. 10 Bandung 40132, Indonesia Hilda Assiyatun Combinatorial Mathematics Research Group Bandung Institute of Technology Jalan Ganesha No. 10 Bandung 40132, Indonesia Saladin Uttunggadewa Combinatorial Mathematics Research Group Bandung Institute of Technology Jalan Ganesha No. 10 Bandung 40132, Indonesia Copyright c 2014 I Wayan Sudarsana, Edy Tri Baskoro, Hilda Assiyatun and Saladin Uttunggadewa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let H be a graph with the chromatic number χ(h) and the chromatic surplus σ(h). A connected graph G of order n is called good with respect to H, denoted by H-good, ifr(g, H) =(n 1)(χ(H) 1)+σ(H). In this paper, we investigate the Ramsey numbers for a union of graphs not necessarily containing an H-good component.

2 768 I W. Sudarsana, E. T. Baskoro, H. Assiyatun and S. Uttunggadewa Mathematics Subject Classification: 05C55 Keywords: (G, H)-free, H-good, Ramsey number, union of graphs 1 Introduction We consider that all graphs in this paper are finite, undirected and simple. For graphs G, H such that H is a subgraph of G, we define G H as the graph obtained from G by deleting the vertices of H and all edges incident to them. The order of a graph G, denoted as G, is the number of vertices of G. We denote a path on n vertices by P n, a star on n vertices by S n, a cycle on n vertices by C n, a complete graph on n vertices by K n, and a wheel on n +1 vertices by W n. Let G and H be two graphs, the Ramsey number R(G, H) is the minimum n such that in every coloring of the edges of the complete graph K n with two colors, say red and blue, there is a red copy of G or a blue copy of H. A graph F is called (G, H) free if F contains no subgraph isomorphic to G and its complement F contains no subgraph isomorphic to H. The Ramsey number R(G, H) can be equivalently defined as the smallest natural number n such that no (G, H) free graph on n vertices exists. The Ramsey numbers R(G, H) have been intensively studied since Chvátal and Harary [8] established the general lower bound R(G, H) (c(g) 1)(χ(H) 1) + 1, where c(g) is the order of the largest component of G and χ(h) is the chromatic number of H. Burr [4] proved the general lower bound R(G, H) (n 1)(χ(H) 1) + σ(h), (1) where G is a connected graph of order n and σ(h) is its chromatic surplus, namely, the minimum cardinality of a color class taken over all proper colorings of H with χ(h) colors. Motivated by this inequality, the graph G is said to be good with respect to H, denoted by H good, if equality holds in (1). Otherwise, G is not H-good. For examples, the H-good graphs with σ(h) = 1 are: tree T n is K m -good for n, m 2[7];P n is W m -good for n m 1 3 and m is even [6]; and C n is W m -good for even m and n 5m 1 [16]. Meanwhile, 2 S n is not W 6 -good for n 3 [5]. Other results concerning the H-goodness of graphs with σ(h) = 1 can be found in Radziszowski [11] and Lin et al. [9]. Let G i be a connected graph with the vertex set V i and the edge set E i. The union of graphs, denoted by k i=1 G i, has the vertex set k i=1 V i and the edge set k i=1 E i.ifg 1 G 2... G k F, where F is a connected graph, then we denote the union of graphs by kf. The results concerning Ramsey number R( k i=1 G i,h) where G i is H-good with σ(h) = 1 for each i can be seen in [1], [2], [10], [12]. Furthermore, in [3], [13], [14] the Ramsey numbers for

3 On the union of graphs Ramsey numbers 769 a union of H-good graphs in which σ(h) is not necessarily one were obtained. In general, the Ramsey number R( k i=1 G i,h) for a union of graphs containing not all H-good components G i is still open. In this paper, we will investigate such Ramsey numbers and the main result as follows. 2 The Main Result The following theorem deals with the Ramsey number for a union of graphs where the component is not all H-good. Theorem 2.1 Let H be a graph with the chromatic number χ(h) 2 and the chromatic surplus σ(h). Let G k i=1 G i be a union of graphs, where G i is a connected graph of order n i satisfying R(G 1,H) R(G 2,H)... R(G k,h). Then, i 1 R(G, H) max R(G i,h)+ n j }. (2) Furthermore, suppose that the maximum of the right side of (2) is achieved for i 0.IfG i0 is H-good and n 1 n 2... n k σ(h) then i 1 R(G, H) = max R(G i,h)+ n j }. (3) Proof. We prove the first statement by induction on k. For k = 1, the assertion is trivial. We shall show that the assertion holds for k = 2, that is R(G 1 G 2,H) max } R(G 1,H),R(G 2,H)+n 1 = t. Let F 1 be a graph on t vertices and F 1 contains no H. We will show that F 1 contains G 1 G 2. Since t R(G 1,H), it implies that F 1 contains G 1. Therefore, the subgraph F 1 G 1 of F 1 has t n 1 vertices. Since t R(G 2,H)+n 1, it follows that t n 1 R(G 2,H) and so F 1 G 1 contains G 2. Thus we have a G 1 G 2 in F 1. Now assume that the theorem is true for all k 1. We shall show that the theorem is also valid for k. Let t be the maximum of the right side of (2) and F be a graph of order t. Suppose that F contains no H. By induction hypothesis on k, we have that F contains k 1 i=1 G i. Since t R(G k,h)+ k 1 i=1 n i,it follows that F k 1 i=1 G i = t k 1 i=1 n i R(G k,h). This implies that the subgraph F k 1 i=1 G i of F contains G k and hence we obtain k i=1 G i in F. Thus R(G, H) t and so the first assertion holds. To prove the second statement we argue as follows. Let t = R(G i0,h)+b 0 be the maximum of the right side of (2) achieved for i 0 with b 0 = i 0 1 n j.

4 770 I W. Sudarsana, E. T. Baskoro, H. Assiyatun and S. Uttunggadewa Since G i0 is H-good, it implies that t =(n i0 1)(χ(H) 1) + σ(h) +b 0. Now consider the graph F K b0 +n i0 1 (χ(h) 2)K ni0 1 K σ(h) 1. Since n 1 n 2... n k σ(h), it follows that F does not contain at least one component G i of G. ThusF contains no G. Meanwhile, F is a complete χ(h)- partites graph and hence of course χ(f ) = χ(h). Since the smallest partite of F has σ(h) 1 vertices, it follows that F does not contain H. Therefore, F is a (G, H)-free graph of order t 1 and so R(G, H) t. This completes the proof of Theorem 2.1. In particular, if G is a union of graphs with k components and each of them has n vertices then the following corollary holds. Corollary 2.2 Let H be a graph with the chromatic number χ(h) 2 and the chromatic surplus σ(h). Let G k i=1 G i be a union of graphs with G i = n satisfying R(G 1,H) R(G 2,H)... R(G k,h). Then, } R(G, H) max R(G i,h)+(i 1)n. (4) Furthermore, suppose that the maximum of the right side of (4) is achieved for i 0.IfG i0 is H-good and n σ(h) then R(G, H) =(n 1)(χ(H) 1) + σ(h)+(i 0 1)n. (5) Proof. Use Theorem 2.1 by defining G k i=1 G i with G i = n for every i and R(G i0,h)=(n 1)(χ(H) 1)+σ(H). Note that Corollary 2.2 gives R(G, H) =(n 1)(χ(H) 1)+(k 1)n+σ(H), whenever G is a union of graphs with kh-good components and each of them has n vertices. Consequently, we have the following corollary which is similar with the results proposed by Bielak [3] and Sudarsana et al. [14]. Corollary 2.3 Let H be a graph with the chromatic number χ(h) 2 and the chromatic surplus σ(h). Let G i be a union of graphs with l i H-good components and each of them has n i vertices. If G k i=1 G i and n 1 n 2... n k σ(h) then } R(G, H) = max (n i 1)(χ(H) 2) + i l j n j + σ(h) 1. (6) Proof. Use Theorem 2.1 by defining G k i=1 G i and G i is a union of graphs with l i H-good components and each of them has n i vertices.

5 On the union of graphs Ramsey numbers 771 Remark. Consider graphs G 1 2C 100 P 90 3S 20, G 2 4C 40 2P 10 3S 9 and W 6. Note that the components S 20 of G 1 and S 9 of G 2 are not W 6 -good, respectively. Thus the conditions of Bielak s theorems in [2] and [3] are not satisfied by G 1 and G 2. It can be verified that all assumptions of Theorem 2.1 are satisfied by G 1 and therefore we obtain R(G 1,W 6 ) = 379. Meanwhile the maximum of the right side of (2) is achieved for the component S 9 of G 2 and hence we only obtain the upper bound R(G 2,W 6 ) 137. In general, it is difficult to obtain the exact value of R(G, H) if the maximum of the right side of (2) is not achieved for an H-good component. The following theorem deals with the Ramsey number for a union of graphs containing no H-good components. In particular, a union of stars with different sizes versus wheel of order seven. Theorem 2.4 Let l i be a natural number. If n 1 n 2... n k 3 and G k i=1 l is ni then R(G, W 6 ) = max (n i +1)+ i l j n j }. (7) Proof. Let t =(n i0 +1)+b 0 be the maximum of the right side of (7) achieved for i 0 with b 0 = i 0 l jn j. It can be verified that K b0 1 H is a(g, W 6 )-free graph on t 1 vertices, where H C ni0 +1 for n i0 5 and H 2C 3 for n i0 = 5. Therefore, R(G, W 6 ) t. The upper bound R(G, W 6 ) t follows from (2) in Theorem 2.1. Note that not all Ramsey numbers for a union of graphs equal to the upper bound (2) in Theorem 2.1. To mention an example, refers to the work of [15] showing that R( k i=1 l is ni,w 4 ) is less than the upper bound (2) for even n i 4. Therefore, it is a challenging problem to obtain the exact Ramsey numbers R(G, H) when G is a union of graphs containing no H-good components. Acknowledgements. The first author gratefully acknowledges the Directorate General of Higher Education (DGHE), Indonesian State Ministry of Education and Culture for financial support under grant Penelitian Fundamental: 125.C/un 28.2/PL/2012. References [1] E. T. Baskoro, Hasmawati and H. Assiyatun, The Ramsey number for disjoint unions of trees, Discrete Mathematics, 306 (2006),

6 772 I W. Sudarsana, E. T. Baskoro, H. Assiyatun and S. Uttunggadewa [2] H. Bielak, Ramsey numbers for a disjoint union of some graphs, Applied Mathematics Letter, 22 (2009), [3] H. Bielak, Ramsey numbers for a disjoint union of good graphs, Discete Mathematics, 310 (2010), [4] S. A. Burr, Ramsey numbers involving long suspended paths, Journal of London Mathematical Society, 24:2 (1981), [5] Y. Chen, Y. Zhang and K. Zhang, The Ramsey number of stars versus wheels, European Journal of Combinatoric 25 (2004), [6] Y. Chen, Y. Zhang and K. Zhang, The Ramsey number paths versus wheels, Discrete Mathematics, 290 (2005), [7] V. Chvátal, Tree complete graphs Ramsey number, Journal of Graph Theory, 1 (1977), 93. [8] V. Chvátal and F. Harary, Generalized Ramsey theory for graphs, III: small off-diagonal numbers, Pacific Journal of Mathematics, 41 (1972), [9] Q. Lin, Y. Li and L. Dong, Ramsey goodness and generalized stars, European Journal of Combinatoric, 31 (2010), [10] Hasmawati, E. T Baskoro and H. Assiyatun, The Ramsey number for disjoint unions of graphs, Discrete Mathematics, 308 (2008), [11] S. P. Radziszowski, Small Ramsey numbers, Electronic Journal of Combinatoric, DS12, August 4, 2009, [12] S. Stahl, On the Ramsey number R(F, K m ) where F is a forest, Canadian Journal of Mathematics 27 (1975), [13] I W. Sudarsana, Adiwijaya and S. Musdalifah, The Ramsey number for a linear forest versus two identical copies of complete graphs, In: Thai, M. T. and Sahni, S. (eds.) COCOON 2010, LNCS 6196, , Springer, Heidelberg (2010). [14] I W. Sudarsana, E. T. Baskoro, H. Assiyatun and S. Uttunggadewa, The Ramsey numbers for the union graph with H-good components, Far East Journal of Mathematical Sciences, 39:1 (2010), [15] I W. Sudarsana, E. T. Baskoro, H. Assiyatun and S. Uttunggadewa, The Ramsey numbers of certain forest respect to small wheels, the Journal of Combinatorial Mathematics and Combinatorial Computing, 71 (2009),

7 On the union of graphs Ramsey numbers 773 [16] Surahmat, E. T. Baskoro and I. Tomescu, The Ramsey numbers of large cycles versus wheels, Descrete Mathematics, 306 (2006), Received: November 9, 2013