On Ramsey Minimal Graphs for the Pair Paths - Science Direct

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ScienceDirect Procedia Computer Science 74 (2015) 15 – 20

International Conference on Graph Theory and Information Security

On Ramsey Minimal Graphs for the Pair Paths Desi Rahmadani, Edy Tri Baskoro, Hilda Assiyatun Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung, Jalan Ganesa 10, Bandung 40132, Indonesia

Abstract For any graphs G and H, we write F → (G, H) to means that in any red-blue coloring of all the edges of F, the graph F will contain either a red G or a blue H. A graph F is called a Ramsey (G,H)-minimal graph if F satisfies two conditions: F → (G, H), and F ∗ (G, H) for every subgraph F ∗ of F. The set of all Ramsey (G, H)-minimal graphs is denoted by R(G, H). In this paper, we construct some family of graphs which belong to R(P3 , Pn ), for any n ≥ 6. In particular, we give an infinite class of trees which provides Ramsey (P3 , P7 )-minimal graphs. 2015Published The Authors. Published by Elsevier B.V. ©c 2015 by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICGTIS 2015. Peer-review under responsibility of the Organizing Committee of ICGTIS 2015

Keywords: Ramsey minimal graph, path, tree. 2010 MSC: 05D10, 05C55

1. Introduction We call a graph F as a Ramsey (G,H)-minimal graph if F →(G, H) but F ∗ (G, H) for every F ∗ ⊂ F. A red-blue coloring of the edges of F is called to be (G, H)-coloring if such a coloring gives neither a red G nor a blue H. The set of all Ramsey (G, H)-minimal graphs is denoted by R(G, H). In particular, the set of all trees in R(G, H) is denoted by RT (G, H). Finding all the Ramsey (G, H)-minimal graphs for particular G and H is a very interesting but difficult problem, even though for small graphs G and H. The pair (G, H) is said to be Ramsey-infinite or Ramsey-finite if the set R(G, H) is infinite or finite, respectively. For instances, the pair (P3 , P3 ) is Ramsey infinite since R(G, H) contains all cycles of odd length. The problem of characterizing pairs of graphs (G, H) that are Ramsey-infinite was first addressed by Neˇsetˇ(r)il and Rödl [15] in 1976. Burr et al. [9] proved that if G is a matching then (G, H) is Ramsey-finite for all graphs H. In 1986, Burr [10] proved that if G is a 2-connected graph, then the pair (G, G) is Ramsey infinite. In 1991, Faudree basically [7] characterized all Ramsey-infinite pairs consisting of two forests. In 1994, Luczak [14] proved that if G is a forest other that a matching and H contains a cycle, then (G, H) is a Ramsey-infinite pair. Then, Rödl and Rudcińskin, they deduced that the pair (G, G) is Ramsey-infinite for all G containing a cycle. E-mail address: [email protected], [email protected], [email protected]

1877-0509 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICGTIS 2015 doi:10.1016/j.procs.2015.12.068

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Desi Rahmadani et al. / Procedia Computer Science 74 (2015) 15 – 20

Burr et al. [7] proposed the following theorem in 1982. Theorem 1. Let T n be a tree on n vertices which is not a star. Then (S k , T n ) is Ramsey infinite if only if k ≥ 2. Yulianti et al. [17] derived some class of graphs belonging to R(P3 , P4 ). Faudree and Sheehan [13] defined a restricted infinite class RT (G, H). They determined all trees in RT (P3 , P4 ) and RT (P3 , P5 ). Motivated by this, we study Ramseyminimal graphs in the infinite class R(P3 , Pn ) for n ≥ 6. 2. Main Results In this section, we give some class of Ramsey (P3 , Pn )-minimal graphs for all n ≥ 6. Let us define the graph F(m), for n ≥ 6 and m = n − 3, as in Figure 1. Then, we have the following theorem.

Fig. 1. Graph F(m)

Theorem 2. The graph F(m) ∈ R(P3 , Pn ), for n ≥ 6 and m = n − 3. Proof. First, we prove that F(m) → (P3 , Pn ). Consider any red-blue coloring of all edges in F(m). Suppose that there is no red P3 , then all of red edges form a matching in F(m). Since, every K3 := (vi , ui+1 , ui+2 ) in F(m) only contain at most one red edge, then there will be a blue path Pn which starts from vertex either u1 or v1 and ends at vm or um+3 . Next, we prove that F(m) − e (P3 , Pn ), for any e in F(m). Let e be in a K3 of F(m). If e = ut ut+1 for some t then color all edges vi ui+2 by red and the remaining edges by blue. By this coloring, there is no red P3 and no blue Pn . If e = ut+1 vt for some t then color edge ut+1 ut+2 and all edges vi ui+2 (for i t) by red and the remaining edges by blue. By this coloring, there is no red P3 and no blue Pn . If e = ut+2 vt for some t then color edge ut+1 ut+2 and all edges vi ui+1 (for i t) by red and the remaining edges by blue. By this coloring, there is no red P3 and no blue Pn in F(m) − e. Therefore, F(m) − e (P3 , Pn ), for any edge e in F(m). Now, consider the graph L(s) as in Fig. 2, for n ≥ 6 and s = n − 5. We will show that this graph belongs to R(P3 , Pn ).

Fig. 2. Graph L(s)

Theorem 3. The graph L(s) ∈ R(P3 , Pn ), for n ≥ 6 and s = n − 5. Proof. First, we prove that L(s) → (P3 , Pn ), n ≥ 6 and s = n − 5. Consider any red-blue coloring of all edges in L(s). Suppose that there is no red P3 , then all of red edges form a matching in L(s). Consider the subgraph A induced by V(L(s)) − {a1 , a2 , · · · , a6 }. Then, there will be a blue path Y on at least s + 2 vertices, starting from u1 to either a7 or v s . Now, consider the subgraph induced by {a1 , a2 , · · · , a6 , u1 }. Then, there will be a path on 4 vertices ending to u1 . Therefore, this path together with the path Y will form a path on at least s + 5 vertices. To prove the minimality, we can use a similar way as in the proof of Theorem 2. Therefore, we have L(s) ∈ R(P3 , Pn ), for n ≥ 6 and s = n − 5.


Next, consider the graph D(s) in Fig. 3, for n ≥ 6 and s = n − 5. We will show that this graph belongs to R(P3 , Pn ).

Fig. 3. Graph D(s)

Theorem 4. The graph D(s) ∈ R(P3 , Pn ), for n ≥ 6 and s = n − 5. Proof. First, we prove that D(s) → (P3 , Pn ). Consider any red-blue coloring of all edges in D(s). Suppose that there is no red P3 , then all of red edges form a matching in D(s). Now, consider the subgraph A induced by V(D(s)) − {a1 , a2 , · · · , a7 }. Then, there will be at most s + 1 red edges in A. Consequently, there will be a blue path on at least s + 2 vertices, starting from u1 . Next, consider the subgraph induced by {a1 , a2 , · · · , a7 , u1 }. Then, there will be a blue path on 4 vertices starting from u1 . These two paths form a path on at least s + 5 vertices. To show the minimality we use a similar way as in the proof of Theorem 2. Therefore, we have D(s) ∈ R(P3 , Pn ), for n ≥ 6 and s = n − 5. In the following theorem, consider the graph B(s) Fig. 4, for n ≥ 6 and s = n − 5. We will show that this graph belongs to R(P3 , Pn ).

Fig. 4. Graph B(s)

Theorem 5. The graph B(s) ∈ R(P3 , Pn ), for n ≥ 6 and s = n − 5. Proof. First, we prove that B(s) → (P3 , Pn ). Consider any red-blue coloring of all edges in B(s). Suppose that there is no red P3 , then all of red edges form a matching in B(s). Now, consider the subgraph A induced by V(B(s)) − {a1 , a2 , . . . , a7 }. Then, there will be at most s + 1 red edges in A. Consequently, there will a blue path on at least s + 2 vertices, starting from u1 . Next, consider the subgraph induced by {a1 , a2 , . . . , a7 , u1 }. Then, there will be a blue path on 4 vertices starting from u1 . These two paths form a path on at least s + 5 vertices. To show the minimality, we use a similar way as in the proof of Theorem 2. Therefore, we have B(s) ∈ R(P3 , Pn ), for n ≥ 6 and s = n − 5. In the following theorem, consider the graph Q(s) in Fig. 5, for n ≥ 6 and s = n − 5. We will show that this graph belongs to R(P3 , Pn ). Theorem 6. The graph Q(s) ∈ R(P3 , Pn ), for n ≥ 6 and s = n − 5. Proof. First, we prove that Q(s) → (P3 , Pn ). Consider any red-blue coloring of all edges in Q(s). Suppose that there is no red P3 , then all of red edges form a matching in Q(s). Now, consider the subgraph A induced by V(Q(s)) − {ai , a2 , . . . , a6 }. Then, there will be at most s + 1 red edges in A. Consequently, there will a blue path on at least s + 2

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Fig. 5. Graph Q(s)

vertices, starting from u1 . Next, consider the subgraph induced by {a1 , a2 , . . . , a6 , u1 }. Then, there will be a blue path on 4 vertices starting from u1 . These two paths form a path on at least s + 5 vertices. To show the minimality, we use a similar way as in the proof of Theorem 2. Therefore, we have Q(s) ∈ R(P3 , Pn ), for n ≥ 6 and s = n − 5. Next, we provide an infinite class of trees in RT (P3 , P7 ). Consider the graph J in Fig. 6 and H1 in Fig. 7, with the path a1 a2 . . . a5 is called the backbone of H1 .

Fig. 6. Graph J

Fig. 7. Graph H1

Theorem 7. The graph H1 ∈ RT (P3 , P7 ). Proof. We first prove that H1 → (P3 , P7 ). Consider any red-blue coloring of all edges of H1 . Suppose that there is no red P3 . Then, consider these three edges a2 a3 , a3 a4 , and a3 a30 . At most one of these edges is red. Then, in any case, H1 will contain a blue path P7 . Hence, H1 →(P3 , P7 ). Second, to prove the minimality of H1 , we can use a similar way as in Theorem 2. Therefore H1 − e (P3 , P7 ) for any edge e. Now, consider the graph H2 as in Fig. 8. Theorem 8. The graph H2 ∈ RT (P3 , P7 ). Proof. We first prove that H2 → (P3 , P7 ). Consider any red-blue coloring of the edges of H2 containing no red P3 . Then, one of these two edges a3 a4 and a4 a5 , say x, will be red. Since otherwise, this blue P3 can be extended into a


Fig. 8. Graph H2

blue P7 is any coloring. The proof is complete. Now, consider the component of H1 − {x} containing a4 . In any case, this component will contain a blue path P7 . Hence, H2 → (P3 , P7 ). Second, to prove the minimality of H2 , we can use a similar way as in Theorem 2. Therefore H2 − e (P3 , P7 ) for any edge e. Definition 9. For i ≥ 3, the graph Hi is defined as a tree in Fig. 9 which contains a backbone path P of order 2i + 3 (P ≡ a1 a2 . . . a2i+3 ) and subtrees C1 , C2 , . . . , Ci−2 , where Ct K1 if t is odd and Ct J if t is even.

Fig. 9. Graph Hi where Ci K1 for i is odd and Ci J for i is even.

Theorem 10. The graph Hi ∈ RT (P3 , P7 ), for i ≥ 3. Proof. By using a similar method, we can verify that graph Hi ∈ RT (P3 , P7 ), for any i ≥ 3. Acknowledgement. This research was supported by Research Grant ”Program Hibah PMDSU ITB-DIKTI”, Ministry of Research, Technology and Higher Education, Indonesia. References 1. Baskoro ET, Nuraeni Y, Ngurah AAG. Upper bound for The size Ramsey numbers for P3 versus C3t or Pn . Journal of Prime Research in Mathematics 2006;2:141-146. 2. Baskoro ET, Vetrik T, Yulianti L. A note on Ramsey (K1,2 , C4 )-minimal graphs of diameter 2. Proceeding of the International Conference 70 years of FCE STU, Bratislava, Slovakia 2000;2:1-4. 3. Baskoro ET, Yulianti L, Assiyatun H. Ramsey (K1,2 , P4 )-minimal graphs of diameter 2. Journal of Comb.Mathematics and Comb. Computing 2008;65:79-90. 4. Borowiecki M, Haluszezak M, Sidorowicz E. On Ramsey minimal graphs. Discrete Mathematics 2004;286:37-43. 5. Borowiecki M, Schiermeyer L, Sidorowicz E. Ramsey (K1,2 , K3 )-minimal graphs. Electronic Journal of Combonatorics 2005;12:R20. 6. Burr SA, Erd˝os P, Faudree RJ, Rosseu CC, Schelp RH. Ramsey-minimal graphs for the pair star, connected graphs. Discrete Mathemathics 1984;38:23-32. 7. Burr SA, Erd˝os P, Faudree RJ, Rosseu CC, Schelp RH. Ramsey-minimal graphs for forests. Studia Scient. Math. Hungar 1982;15:265-273. 8. Burr SA, Erd˝os P, Faudree RJ, Rosseu CC, Schelp RH. Ramsey-minimal graphs for matchings. The theory and applications of graphs (Kalamazoo, Mich., 1980), Wiley, New York 1981;159-168. 9. Burr SA, Erd˝os P, Faudree RJ, Rosseu CC, Schelp RH. A class of Ramsey finite graphs. Proceedings of the Ninth Southeastern Conference of Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla, Winnipeg, Man.), Utilitas Math 1978;171-180.

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