On the Ramsey-Goodness of Paths

Transcription

1 Graphs and Combinatorics (016) 3: DOI /s z ORIGINAL PAPER On the Ramsey-Goodness of Paths Binlong Li 1, Halina Bielak 3 Received: 8 July 015 / Revised: 1 February 016 / Published online: 6 June 016 The Author(s) 016. This article is published with open access at Springerlink.com Abstract For a graph G, we denote by ν(g) the order of G, byχ(g) the chromatic number of G and by σ(g) the minimum size of a color class over all proper χ(g)- colorings of G. For two graphs G 1 and G, the Ramsey number R(G 1, G ) is the least integer r such that for every graph G on r vertices, either G contains a G 1 or G contains a G. Suppose that G 1 is connected. We say that G 1 is G -good if R(G 1, G ) = (χ(g ) 1)(ν(G 1 ) 1) + σ(g ). In this note, we obtain a condition for graphs H such that a path is H-good. Keywords Ramsey number Goodness Path Mathematics Subject Classification 05C55 05D10 1 Introduction Throughout this paper, all graphs are finite and simple. Let G 1 and G be two graphs. The Ramsey number R(G 1, G ), is defined as the least integer r such that for every The first author is supported by NSFC (No ) and the project NEXLIZ CZ.1.07/.3.00/ B Halina Bielak hbiel@hektor.umcs.lublin.pl Binlong Li libinlong@mail.nwpu.edu.cn 1 Department of Applied Mathematics, Northwestern Polytechnical University, Xi an, Shaanxi 71007, People s Republic of China European Centre of Excellence NTIS, University of West Bohemia, Pilsen, Czech Republic 3 Institute of Mathematics, Maria Curie-Skłodowska University, Lublin, Poland

2 54 Graphs and Combinatorics (016) 3: graph G on r vertices, either G contains a G 1 or G contains a G, where G is the complement of G. We denote by ν(g) the order of G, byδ(g) the minimum degree of G, byω(g) the component number of G, byχ(g) the chromatic number of G and by σ(g) the minimum size of a color class over all proper χ(g)-colorings of G. Fortwo disjoint graphs G 1 and G,theunion of G 1 and G is defined as V (G 1 G ) = V (G 1 ) V (G ) and E(G 1 G ) = E(G 1 ) E(G ); and the join of G 1 and G is defined as V (G 1 G ) = V (G 1 ) V (G ), and E(G 1 G ) = E(G 1 + G ) {xy : x V (G 1 ), y V (G )}. The union of k disjoint copies of the same graph G is denoted by kg. Theorem 1 (Burr [5]) For all graphs G and H, with G connected and ν(g) σ(h), R(G, H) (χ(h) 1)(ν(G) 1) + σ(h). We say G is H-good if R(G, H) = (χ(h) 1)(ν(G) 1) + σ(h). Lin et al. proved the following theorem on Ramsey-goodness of trees. Theorem (Lin et al. [11]) Let T be a tree and H be a graph. If T is H-good and σ(h) = 1, then T is also K 1 H-good. By P n and C n we denote the path and cycle on n vertices, respectively. For the case of T being a path, we get an extension of Theorem. Theorem 3 Let n, and H be a subgraph of K a1,a,...,a k,k= χ(h), such that k(n 1) + 1 a i, 1 i k, k + i If P n is H-good, then P n is also K 1 H-good. Note that H is a subgraph of K a1,a,...,a k if and only if there is a proper coloring of H such that the size of the ith color class is at most a i,1 i k. We prove Theorem 3 in Sect. 3. In Sect., we will apply Theorem 3 to show some results of Ramsey values involving paths. Note that σ(k 1 H) = 1. By Theorem, we can see that under the conditions of Theorems and 3, P n is K t H-good for all t 1. Theorem 4 Let n 3 and H be a graph with σ(h).ifp n is H-good, then P n is also (K 1 H)-good. Proof Since P n is H-good, R(P n, H) = (χ(h) 1)(n 1) + σ(h). Set r = (χ(k 1 H) 1)(n 1) + σ(k 1 H) = χ(h)(n 1) + σ(h). Then r = R(P n, H) + n 1. From Theorem 1,wehaveR(P n, K 1 H) r.nowletg be an arbitrary graph of order r without P n as a subgraph. We will prove that G contains K 1 H. Let P = v 1 v...v k be a longest path of G. Thus k n 1.

3 Graphs and Combinatorics (016) 3: If k = 1 then G is an empty graph and G is complete. Note that ν(g) = r R(P n, H) + ν(h) +. So G contains K 1 H. Now we assume that k n 1. Let G be a subgraph of G induced by V (G) V (P). Then ν(g ) = r k R(P n, H).SoG contains H. Note that v 1 and v k are nonadjacent to every vertex of G. Thus G contains K 1 H. From Theorem 4 we get the following result. Corollary 1 Let n 3 and H be a graph with σ(h). IfP n is H-good, then P n is also (tk H)-good. Some Corollaries In this section, we will list some known results for the Ramsey numbers involving paths. After each result, we apply Theorems, 3 and 4 to get a new Ramsey numbers involving paths. We denote by L t s (s t + 1) the graph obtained from K s+t by removing the edges of a matching of size t, i.e., L t s = tk K s t. We use par(m) to denote the parity of m. In the following corollaries, we always assume that n 3. Theorem 5 (Gerencsér and Gyárfás [10]) If m n, then m R(P n, P m ) = n + 1. Corollary Let t 0 be an integer. If m n, then R(P n, L t s P m) = (s + 1)(n 1) + 1. Proof By Theorem 5, P n is P m -good. Take a 1 = m/ and a = m/. By Theorem 3, P n is (K 1 P m )-good. So by Theorems and 4 we get the assertion. The kipas K m is the graph obtained by joining a K 1 and a path P m. For the case (s, t) = (1, 0) in Corollary, we can get the values of path-kipas Ramsey numbers R(P n, K m ) for 3 m n, which was already obtained by Saleman and Broersma [16]. Theorem 6 (Faudree et al. [9]) If n and m 3, then n 1, n m and m is odd; n + m/ 1, n m and m is even; R(P n, C m ) = max{m + n/ 1, n 1}, m > n and m is odd; m + n/ 1, m > n and m is even. Corollary 3 Let t 0 be an integer. If m is even, 4 m n; or m is odd and 3 m 3n/, then R(P n, L t s C m) = (s + par(m) + 1)(n 1) + 1.

4 544 Graphs and Combinatorics (016) 3: Proof From Theorem 6, one can check that P n is C m -good. For the case m is even, take a 1 = a = m/ and apply Theorems 3, and 4; for the case m is odd, apply Theorems and 4. In both case, we have the assertion. The wheel W m is the graph obtained by joining K 1 and a cycle C m. For the case (s, t) = (1, 0) in Corollary 3, we can get the values of path-wheel Ramsey numbers R(P n, C m ) under the condition of Corollary 3, which was already obtained by Chen et al. [6]. Theorem 7 If n, then R(P n, mk 1 ) = m. This theorem is trivial and the following corollary can be get immediately. We omit the proof. Corollary 4 Let t 0 be an integer. If m n/, then R(P n, L t s mk 1) = s(n 1) + 1. For m, the graph K 1,m is called a star; the graph K mk 1 is called a book; and the graph K t mk 1, t 3, is called a generalized book. We remark here that the Ramsey numbers of paths versus stars and paths versus (generalized) books under the condition of Corollary 4 was already obtained by Parsons [1], and Rousseau and Sheehan [14], respectively. Theorem 8 (Faudree and Schelp [8]) If n, m i, 1 i k, then ( ) { } k k mi k n R P n, P mi = max n + 1, m i + 1. Corollary 5 Let t 0 be an integer. If m i, 1 i k and k m i / n/, then ( ) k R P n, L t s P mi = (s + 1)(n 1) + 1. Proof By Theorem 8, P n is ( k P mi )-good. Take a 1 = k mi and a = k mi. By Theorem 3, P n is (K 1 k P mi )-good. By Theorems and 4 we get the assertion.

5 Graphs and Combinatorics (016) 3: The graph F m = K 1 mk is called a fan. From the above corollary, we can see that if m n/, then R(P n, F m ) = n 1. This result was already obtained by Saleman and Broersma [15]. Let P k n be the k-th power of P n, i.e., the graph with vertex set {v 1,...,v n } and edge set {v i v j : i j k}. Theorem 9 (Pokrovskiy [13]) If n k + 1, then n R(P n, Pn k ) = k(n 1) +. k + 1 Corollary 6 Let t 0 be an integer. If n k + 1, then R(P n, L t s Pk n ) = (t + k)(n 1) + 1. Proof Note that χ(pn k) = k + 1 and σ(pk n ) = n/(k + 1). By Theorem 9, P n is Pn k -good. Take n + i 1 a i =, 1 i k + 1. k + 1 It is easy to see that Pn k is a subgraph of K a 1,a,...,a k+1. By Theorems 3, and 4, we have the assertion. Theorem 10 (Sudarsana et al. [18]) If m, then R(P n, K m ) = (m 1)(n 1) +. Corollary 7 Let t 0 be an integer. If m and n 3, then R(P n, L t s K m) = (s + m 1)(n 1) + 1. Proof By Theorem 10, P n is K m -good. Take a i =, 1 i m. Note that K m is a subgraph of K a1,a,...,a m. By Theorems 3, and 4, we have the assertion. Theorem 11 (Sudarsana [17]) If m, k and n (k )((km )(m 1)+1)+3, then R(P n, kk m ) = (m 1)(n 1) + k. Corollary 8 Let t 0 be an integer. If m, k and n (k )((km )(m 1) + 1) + 3, then R(P n, L t s kk m) = (s + m 1)(n 1) + 1. Proof By Theorem 11, P n is kk m -good. Take a i = k, 1 i m. Note that kk m is a subgraph of K a1,a,...,a m. By Theorems 3, and 4, we have the assertion.

6 546 Graphs and Combinatorics (016) 3: The cocktail party graph (or hyperoctahedral graph) H m is the graph obtained by removing a perfect matching from a complete graph K m (i.e., H m = mk ). Theorem 1 (Ali et al. [1]) If n, m 3, then R(P n, H m ) = (n 1)(m 1) +. Corollary 9 Let t 0 be an integer. If n, m 3, then R(P n, L t s H m) = (s + m 1)(n 1) + 1. Proof By Theorem 1, P n is H m -good. Take a i =, 1 i m. Note that H m = K a1,a,...,a m. By Theorems 3, and 4, we have the assertion. The sunflower graph SF m is the graph on m+1 vertices obtained by taking a wheel W m with hub x,anm-cycle v 1 v v m v 1, and additional m vertices w 1,w,...,w m, where w i is joined by edges to v i,v i+1,1 i m, where v m+1 = v 1. Theorem 13 (Ali et al. [4]) If m 3, then R(P n, SF m ) = { n + m/, m is even and n 4m 7m + 4; 3n, m is odd and n m 9m Corollary 10 Let t 0 be an integer. If m 4 is even and n 4m 7m + 4, or m 3 is odd and n m 9m + 11, then R(P n, L t s SF m) = (s + + par(m))(n 1) + 1. Proof By Theorem 13, P n is SF m -good. If m is even, then take a 1 = m + 1 and a = a 3 = m/; if m is odd, then σ(sf m ) = 1. By Theorems 3, and 4,wehavethe assertion. The Beaded wheel BW m is a graph on m+1 vertices which is obtained by inserting one vertex in each spoke of the wheel W m. Theorem 14 (Ali et al. [3]) If m 3, then R(P n, BW m ) = { n 1 m is even and n m 5m + 4; n m is odd and n m 5m + 3. Corollary 11 Let t 0 be an integer. If m 4 is even and n m 5m + 4, or m 3 is odd and n m 5m + 3, then R(P n, L t s BW m) = (s + )(n 1) + 1. Proof By Theorem 14, P n is BW m -good. If m is even, then σ(bw m ) = 1; if m is odd, then take a 1 = m and a = a 3 = (m + 1)/. By Theorems 3, and 4, wehave the assertion.

7 Graphs and Combinatorics (016) 3: The Jahangir graph J m is a graph on m + 1 vertices consisting of a cycle C m with one additional vertex which is adjacent alternatively to m vertices of C m. Theorem 15 (Surahmat and Tomescu [19]) If m and n (4m 1)(m 1)+1, then R(P n, J m ) = n + m 1. Corollary 1 Let t 0 be an integer. If m and n (4m 1)(m 1) + 1, then R(P n, L t s J m) = (t + 1)(n 1) + 1. Proof By Theorem 15, P n is J m -good. Take a 1 = m and a = m + 1. By Theorems 3, and 4, we have the assertion. The generalized Jahangir graph J k,m is a graph on km + 1 vertices consisting of a cycle C km with one additional vertex which is adjacent to m vertices of the C km each of which is at distance k to the next one on C km. Theorem 16 (Ali et al. []) If m, k, then n + km/ 1, k is even and n (km 1)(km/ 1) + 1; R(P n, J k,m ) = n 1 k is odd, m is even and n km(km )/; n k, m are odd and n (km 1) /. Corollary 13 Let t 0 be an integer. If n, m, k, and if k is even and n (km 1)(km/ 1) + 1, or k is odd, m is even and n km(km )/, ork, m are odd and n (km 1) /, then R(P n, L t s J k,m) = (s par(k))(n 1) + 1. Proof By Theorem 16, P n is J k,m -good. If k is even, then take a 1 = km/ + 1 and a = km/; if k is odd, then take k + k + 1 a 1 = m + 1, a = m and a 3 = m 3 3 k. 3 By Theorems 3, and 4, we have the assertion. 3 Proof of Theorem 3 From Theorem 1, it is sufficient to prove that R(P n, K 1 H) k(n 1) + 1. Let G be a graph of order k(n 1) + 1. Suppose that G contains no P n and G contains no K 1 H. Since H is a subgraph of K a1,a,...,a k,wehave k(n 1) + 1 σ(h) a k = k n k 1 n =. k

8 548 Graphs and Combinatorics (016) 3: Since P n is H-good, n R(P n, H) = (χ(h) 1)(n 1) + σ(h) (k 1)(n 1) +. If there is a vertex v in G with d(v) n/ 1, then let G be a subgraph of G induced by V (G) {v} N(v), where N(v) is the set of vertices adjacent to v in G. Note that n ν(g ) = ν(g) 1 d(v) k(n 1) + 1 n = (k 1)(n 1) + R(P n, H). This implies that G contains a path P n or G contains a subgraph isomorphic to H. Note that v is nonadjacent to every vertex of G. G contains a P n or G contains a K 1 H, a contradiction. Thus we assume that δ(g) n/. If there is a component B of G with ν(b) n, then by Dirac s Theorem (see [7]), B contains a P n, a contradiction. Thus we assume that every component of G has order at most n 1. Note that the minimum degree of G is at least n/. Every component of G has order between n/ +1 and n 1. If ω(g) k, then ν(g) k(n 1); and if ω(g) k, then ν(g) k(n + 1), both a contradiction. This implies that k + 1 ω(g) k 1. Let B ={B 1, B,...,B ω }, ω = ω(g), be the set of the components of G. We assume without loss of generality that ν(b 1 ) ν(b ) ν(b ω ). Thus we have ν(g) (i 1)(n 1) (k i + 1)(n 1) + 1 ν(b i ) =, 1 i k <ω. ω i + 1 ω i + 1 Now we partition B into k + 1 parts such that the order sum of the components in the ith part is at least a i,1 i k. Let t = ω k 1. For 1 i t, letb i ={B ω i+1, B ω i };fort + 1 i k, let B i ={B i t }; and let B k+1 ={B ω }. If 1 i t, then B i contains two components each of which has order at least n/ +1. Thus {ν(b j ) : B j B i } n + 1. On the other hand, k(n 1) + 1 k(n 1) + 1 a i = n < ν(b j ). k + i k B j B i If t + 1 i k, then B i ={B i t }. Note that (k i + t + 1)(n 1) + 1 (ω i)(n 1) + 1 ν(b i t ) =. ω i + t + 1 ω k i

9 Graphs and Combinatorics (016) 3: Since ω k i k, one can check that k(n 1) + 1 (ω i)(n 1) + 1 a i ν(b i t ). k + i ω k i Clearly ν(b ω ) 1. Thus G contains a K a1,a,...,a k,1, which is a supergraph of K 1 H, our final contradiction. The proof is complete. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. References 1. Ali, K., Baig, A.Q., Baskoro, E.T.: On the Ramsey number for a linear forest versus a cocktail party graph. J. Comb. Math. Comb. Comput. 71, (009). Ali, K., Baskoro, E.T., Tomescu, I.: On the Ramsey numbers for paths and generalized Jahangir graphs J s,m. Bull. Math. Soc. Sci. Math. Roumanie Tome 51(3), (008) 3. Ali, K., Baskoro, E.T., Tomescu, I.: On the Ramsey number for paths and beaded wheels. J. Prime Res. Math. 5, (009) 4. Ali, K., Tomescu, I., Javaid, I.: On path-sunflower Ramsey numbers. Math. Rep. 17(4), (015) 5. Burr, S.A.: Ramsey numbers involving graphs with long suspended paths. J. Lond. Math. Soc. 4, (1981) 6. Chen, Y., Zhang, Y., Zhang, K.: The Ramsey numbers of paths versus wheels. Discret. Math. 90(1), (005) 7. Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc., (195) 8. Faudree, R.J., Schelp, R.H.: Ramsey numbers for all linear forests. Discret. Math. 16(), (1976) 9. Faudree, R.J., Lawrence, S.L., Parsons, T.D., Schelp, R.H.: Path-cycle Ramsey numbers. Discret. Math. 10(), (1974) 10. Gerencsér, L., Gyárfás, A.: On Ramsey-type problems. Ann. Univ. Sci. Budapest Eötvös Sec. Math. 10, (1967) 11. Lin, Q., Li, Y., Dong, L.: Ramsey goodness and generalized stars. Eur. J. Comb. 31(5), 18 4 (010) 1. Parsons, T.D.: Path-star Ramsey numbers. J. Combin. Theory Ser. B 17(1), (1974) 13. Pokrovskiy, A.: Calculating Ramsey numbers by partitioning coloured graphs, eprint arxiv: Rousseau, C.C., Sheehan, J.: A class of Ramsey problems involving trees. J. Lond. Math. Soc. 18, (1978) 15. Salman, A.N.M., Broersma, H.J.: Path-fan Ramsey numbers. Discret. Appl. Math. 154(9), (006) 16. Salman, A.N.M., Broersma, H.J.: Path-kipas Ramsey numbers. Discret. Appl. Math. 155(14), (007) 17. Sudarsana, I.W.: The goodness of long path with respect to multiple copies of complete graphs. J. Indones. Math. Soc. 0(1), (014) 18. Sudarsana, I.W., Assiyatun, H., Adiwijaya, Musdalifah, S.: The Ramsey number for a linear forest versus two identical copies of complete graphs. Discret. Math. Alg. Appl. (4), (010) 19. Surahmat, Tomescu, I.: On path-jahangir Ramsey numbers. Appl. Math. Sci. 8(99), (014)