Unicyclic Ramsey (mk 2, P n )-Minimal Graphs
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1 Unicyclic Ramsey (mk 2, P n )-Minimal Graphs 1 Kristiana Wijaya, 2 Edy Tri Baskoro and 2 Hilda Assiyatun 1 Department of Mathematics, Universitas Jember, 1,2 Combinatorial Mathematics Research Group, Institut Teknologi Bandung, Indonesia September 12, 2013 Graph Theory Conference in honor of Egawa s 60th birthday Kagurazaka Campus of Tokyo University of Science Kristiana W. () Ramsey Minimal September 12, / 66
2 Outlines 1 Introduction 2 Main Results 3 Open Problem Kristiana W. () Ramsey Minimal September 12, / 66
3 Introduction Definition 1 For any given graphs G and H, notation F (G, H) means that in any red-blue coloring on the edges of graph F there exists a red copy of G or a blue copy of H in F. Kristiana W. () Ramsey Minimal September 12, / 66
4 Introduction Definition 1 For any given graphs G and H, notation F (G, H) means that in any red-blue coloring on the edges of graph F there exists a red copy of G or a blue copy of H in F. Example 1 Kristiana W. () Ramsey Minimal September 12, / 66
5 Introduction Definition 1 For any given graphs G and H, notation F (G, H) means that in any red-blue coloring on the edges of graph F there exists a red copy of G or a blue copy of H in F. Example 1 Kristiana W. () Ramsey Minimal September 12, / 66
6 Introduction Definition 2 A (G, H)-coloring is a red-blue coloring where neither a red G nor a blue H occurs. Kristiana W. () Ramsey Minimal September 12, / 66
7 Introduction Definition 2 A (G, H)-coloring is a red-blue coloring where neither a red G nor a blue H occurs. Example 2 Kristiana W. () Ramsey Minimal September 12, / 66
8 Introduction Definition 3 Graph F is Ramsey (G, H) minimal if : (1) F (G, H). (2) F e (G, H) for any edge e F. R(G, H) : The class of all Ramsey (G, H)-minimal graphs. Kristiana W. () Ramsey Minimal September 12, / 66
9 Introduction Some Previous Results Kristiana W. () Ramsey Minimal September 12, / 66
10 Introduction Some Previous Results Kristiana W. () Ramsey Minimal September 12, / 66
11 Introduction Some Previous Results Kristiana W. () Ramsey Minimal September 12, / 66
12 Some Previous Results Lemma 3. [Wijaya et al.] The only forest in R(mK 2, P n ) is mp n. Kristiana W. () Ramsey Minimal September 12, / 66
13 Some Previous Results Lemma 3. [Wijaya et al.] The only forest in R(mK 2, P n ) is mp n. Proof Kristiana W. () Ramsey Minimal September 12, / 66
14 Purpose of our talk To find: 1 all cycle graphs in R(mK 2, P n ) and 2 all (connected) unicyclic graphs in R(mK 2, P 3 ). Kristiana W. () Ramsey Minimal September 12, / 66
15 Lemma 1 If F R(mK 2, P n ) then for every (i) v 1, v 2,, v m 1 V (F), F {v 1, v 2,, v m 1 } P n, (ii) v 1, v 2,, v k V (F) and lf3 in F, F {v 1, v 2,, v k } E(lF3 ) P n, where k, l = 1, 2,, m 2 and k + l = m 1, (iii) (m 1)F 3 in F, F E((m 1)F 3 ) P n, (iv) v 1, v 2,, v k V (F) and F2s+1 in F, F {v 1, v 2,, v k } E(F2s+1 ) P n, where k = 1, 2,, m 2, s = 1, 2,, m 2 and k + s = m, (v) F 2m 1 in F, F E(F 2m 1 ) P n. F k is induced connected subgraph of F with k vertices. Kristiana W. () Ramsey Minimal September 12, / 66
16 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
17 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
18 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
19 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
20 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
21 Lemma 2 If F R(mK 2, P n ) then F {v} contain some Ramsey ((m 1)K 2, P n ) minimal graph for every v V (F). Kristiana W. () Ramsey Minimal September 12, / 66
22 Proof Kristiana W. () Ramsey Minimal September 12, / 66
23 Proof Kristiana W. () Ramsey Minimal September 12, / 66
24 Proof Kristiana W. () Ramsey Minimal September 12, / 66
25 Proof Kristiana W. () Ramsey Minimal September 12, / 66
26 Proof Kristiana W. () Ramsey Minimal September 12, / 66
27 Theorem 1 R(mK 2, P n ) contains no tree. Proof Kristiana W. () Ramsey Minimal September 12, / 66
28 Theorem 1 R(mK 2, P n ) contains no tree. Proof Kristiana W. () Ramsey Minimal September 12, / 66
29 Theorem 1 R(mK 2, P n ) contains no tree. Proof If d(v n 1 ) = 2 then v n 1 v n 2 and v n 1 v n. By Lemma 2, T {v n } Kristiana W. () Ramsey Minimal September 12, / 66
30 Theorem 1 R(mK 2, P n ) contains no tree. Proof If d(v n 1 ) = 2 then v n 1 v n 2 and v n 1 v n. By Lemma 2, T {v n } Kristiana W. () Ramsey Minimal September 12, / 66
31 Theorem 1 R(mK 2, P n ) contains no tree. Proof If d(v n 1 ) = 2 then v n 1 v n 2 and v n 1 v n. By Lemma 2, T {v n } Kristiana W. () Ramsey Minimal September 12, / 66
32 Theorem 1 R(mK 2, P n ) contains no tree. Proof If d(v n 1 ) 3 Kristiana W. () Ramsey Minimal September 12, / 66
33 Theorem 1 R(mK 2, P n ) contains no tree. Proof If d(v n 1 ) 3 then u / V (L) u v n 1. Kristiana W. () Ramsey Minimal September 12, / 66
34 Theorem 1 R(mK 2, P n ) contains no tree. Proof If d(v n 1 ) 3 then u / V (L) u v n 1. By Lemma 2, T {v n 1 } Kristiana W. () Ramsey Minimal September 12, / 66
35 Theorem 1 R(mK 2, P n ) contains no tree. Proof If d(v n 1 ) 3 then u / V (L) u v n 1. By Lemma 2, T {v n 1 } Kristiana W. () Ramsey Minimal September 12, / 66
36 Theorem 1 R(mK 2, P n ) contains no tree. Proof If d(v n 1 ) 3 then u / V (L) u v n 1. By Lemma 2, T {v n 1 } Kristiana W. () Ramsey Minimal September 12, / 66
37 Theorem 2 A cycle graph C s R(mK 2, P n ) if and only if mn n + 1 s mn 1. Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
38 Theorem 2 A cycle graph C s R(mK 2, P n ) if and only if mn n + 1 s mn 1. Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
39 Theorem 2 A cycle graph C s R(mK 2, P n ) if and only if mn n + 1 s mn 1. Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
40 Theorem 2 A cycle graph C s R(mK 2, P n ) if and only if mn n + 1 s mn 1. Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
41 Theorem 2 A cycle graph C s R(mK 2, P n ) if and only if mn n + 1 s mn 1. Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
42 Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
43 Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
44 Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
45 Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
46 Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
47 Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
48 Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
49 Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
50 Proof of Theorem 2 Kristiana W. () Ramsey Minimal September 12, / 66
51 Lemma 4 Let F be a (connected) unicyclic graph other than cycle. If F R(mK 2, P 3 ) then 2m V (F) 3(m 1). Proof By Lemma 1, we have V (F) 2m and we have C 3m 2, C 3m 1 R(mK 2, P 3 ) by Theorem 2. Kristiana W. () Ramsey Minimal September 12, / 66
52 Definition 4 Graph C t s is a graph cycle C s with t pendant vertices such that d(v) = 2 or d(v) = 3 for all v C s. Example Kristiana W. () Ramsey Minimal September 12, / 66
53 Definition 5 Cs t has a gap sequence (a 1, a 2,, a t 1 ) if all pendant vertices attaching to vertices v u1, v u2,, v ut (respectively) satisfy Example d(v ui, v ui+1 ) = a i for i = 1, 2,, t 1. Kristiana W. () Ramsey Minimal September 12, / 66
54 Theorem 3 Let 2m s 3(m 1) and t = 3m 1 s. If C t s has a gap sequence (2, 2,, 2) or (1, 2,, 2) then C t s R(mK 2, P 3 ). Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
55 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
56 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
57 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
58 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
59 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
60 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
61 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
62 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
63 Theorem 4 If C t s R(mK 2, P 3 ) with a gap sequence (a 1, a 2,, a t 1 ), then C t+1 s 1 R(mK 2, P 3 ) with a gap sequence (a 1, a 2,, a t 1, 2). Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
64 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
65 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
66 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
67 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
68 Illustration Proof Kristiana W. () Ramsey Minimal September 12, / 66
69 Open Problem Kristiana W. () Ramsey Minimal September 12, / 66
70 Kristiana W. () Ramsey Minimal September 12, / 66
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