The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN ; APR 3 H996 JAN 6 1997 L161— O-1096 ■r fct I ///a i k d UIUCDCS-R-77-906 UILU-ENG 77 1761 A GENERALIZATION OF RAMSEY THEORY FOR GRAPHS by C. L. Liu and K. M. Chung December 1977 A GENERALIZATION OF RAMSEY THEORY FOR GRAPHS by C. L. Liu and K. M. Chung Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois December 1977 This work was supported in part by the National Science Foundation under grant MCS 73-03408. Digitized by the Internet Archive in 2013 http://archive.org/details/generalizationof906liuc I. INTRODUCTION Let c, d, , d ? , ... , d.be integers such that 1 <_ d, < d ? < . . . - d, < c. Let t denote the sum ( h ) + ( h ) + ' * ' + ( d ) " Glven c distinct colors, we order the( , ) subsets of d, colors, f , j subsets of d^ colors, ... , and ( j j subsets of d. colors in some arbitrary manner. Let G, , G ? , ... , G. be graphs. The (d, , d„, ... , d.) - chromatic Ramsey number for 6,, Gp, ... , G,, denoted R. . , (G,, G 2 , ... , G.), is defined to be the least integer s such that if the edges of the complete graph K are colored in any fashion with c colors, then for some i the subgraph whose edges are colored with the i subset of colors contains a G.. The p case R-,(G-|, G 2 , ... , G ) was studied extensively in the literature, see, p for example, [1]. When G, , G~, ... , G are complete graphs, R-JG, , G~, p ... , G ) is indeed the classical Ramsey numbers [5,6]. The case R ? (G,, G ? ... , G t ) was studied in [2,3]. In this paper, we shall investigate the p case R-, i^i' G?> ... , G.). In particular, we shall determine some 3 Ramsey numbers for complete graphs, R-, 9 (K. , K . , K , K , K , K ) and R^ 3 (K is Kj, K^, K m , K p , K , K , K r ) . Thus, for example, R^ 2 (K r K . , K„, K , K , K ) is the smallest integer s such that when the edges of i m n p the complete graph K are colored in any fashion with 3 colors, denoted a, 6, y> either there is an a K. , or a B K. , or a y K or an a - 3 K or an a - y K , or an B - y K . In other words, s - 1 is the largest integer such that the complete graph K , can be decomposed into three subgraphs H , H„, H with H containing no K. nor an independent set of size p, l-L containing no K. nor an independent set of size n, and H 3 J Y containing no K nor an independent set of size m. II. SOME UPPER AND LOWER BOUNDS We show in this section some upper and lower bounds for R^ ? (K., K., K , K , K , K ). Similar bounds can be obtained for R JK., K. s K , J ill ll p I jO I J K , K , K , K , K ) which we shall omit here. Theorem 2.1 : R? „(K,, K . , K , K , K , K ) < R?(K., K., K ). 1 , 2 v i j i m n p — 1 l j V Furthermore, if m _> R?(K. , K.), n > R^(K, , K ), and p > R^(K., K ) then R? 9 (K,, K., K, K , K , K ) = R?(K., K., K ). 1 , 2 i j v m n p ' 1 v l j ' i ' Proof: That R^ 2 (K., K jS K £ , K m , K p , K p ) < Rj(K r K . , K £ ) is 3 obvious. Suppose R,(K., K., K,) = s. Thus, there is a 3-coloring of the I I J X, edges of K n that contains no a L, nor 3 K., nor y K n . We note that s-1 l j $. 2 if m >_ R-. (K. , K.) this 3-coloring of K , does not contain an u-j K either. (The existence of an a - 6 K implies the existence of either v m v an a K. or a 3 K..) Similarly, if n >. RnK. , K ) and p > R^(K., K ) this I J I I X/ I J X- 3-coloring of K , does not contain an a-y K nor a 3 - y K . D s-1 ' n ' p Theorem 2.2 : R? 9 (K., K. , K n , K , K , K ) < R 9 (K , K , K ). 1,2 V "i j i m n p - 2 X m n p Furthermore, if i >_ min(m,n) , j >_min(m,p), and z >_ min(n,p) then R^ 9 (K., K., K , K , K , K ) = R 9 (K , K , K ). 1 , 2 v l ' j v, m n p ' 2 v m ' n p ' Proof : Obvious. □ Theorem 2.3: R? 9 (K., K. , K n , K , K , K ) < R?(K., K ) 1 ,2 V i j l m n p — 1 i P R? 9 (K., K., K , K , K , K ) < R?(K., K ) 1 ,2 V i j l m n p - 1 J n' R? 9 (K., K., K n , K , K , K ) < R?(K,, K ) 1 ,2 V i j £' m n p - 1 l m' Proof: Obvious. n Theorem 2.4 : Let ,2,„ .. > . „2 t, = mmtRfd^.Kp) ♦ Rf(K j . 1 .K n ) - 2, Ri.zfKi.Kj.K^Vv^.Kp) - l] n[^(S-TS' + R l (K *-l' K m> " 2 ' <2< K i- K j'Wn-rV " in[ R 2(K r K n ) + R?(K,. r g - 2, R^ 2 (K r K K t ,K m ,K n ,K ) - l] to = mil then, R^ 2 (K.,K K ? , K m , K n , K) < ^-(t 1 + t 2 + t 3 + 1) + 1 Proof : Consider an arbitrary 3-coloring of K, for some h. Let N , N , N denote the number of a edges, 3 edges, and y edges incident with any one of the vertices, respectively. If there is no a K., 3 K., YK n ,a-3K,a-yK, and 3 - y K„ in this 3-colorinq of K, the follow- ' £ m r n ' p h ing inequalities must be satisfied simultaneously. N i¥ K i-r V - 1 N 3- R t (K j-l' K n> - 1 N a + N 3- R i, 2 ( K i' K r V Vr V V - 1 N + N < R? 9 (K., K., K n , K , K ,, K ) - 1 a y - 1 >2 X i j V m n-1 p y N + N < R? (K., K., K„, K M , K , K ,) - 1 3 y — i >2 V i j l M n p- V Thus, the following inequalities must be satisfied simultaneously N + N < t, a 3 — I \ + N a y 3 Y — 3 where t, , t^, t 3 are defined as above. In other words, if "1 N + N + N > a 3 Y — (t, + t 9 + t. + 1) 2 v-| then either there is an a K. , or a 6 K. , or a y K , or an a - 8 K , or an a - B K . or a s - y K in any 3-coloring of K. . It follows that R i,2 {i ~ ]) (j ~ ]) Proof : The coloring scheme shown in Fig. 2.1 establishes the lower bound. ° Corollary 2.5.1 : Let T be an arbitrary tree with i vertices and T 1 be an arbitrary tree with j vertices, then R^ 2 (T, K-, T', K., K., K.) = (i - 1 ) ( j - 1 ) + 1 Proof : According to Chvatal [4], R^(T, K.) = (i - 1 ) (j - 1 ) + 1 . + The result in this corollary can, indeed, be considered as a generaliza- tion of Chvatal 's result. 3 III. R 1>2 (K 3> K 3 , K 3 , K m , K n> K p ) In this section, we determine all Ramsey numbers of the form R? ? (K 3 , K v K 3 , K , K , K ). (Clearly, it is only meaningful that m, n, p are larger than or equal to 3.) K 3 , K 3 , K 3 , K 3 , K p ) = 6 ,5 p <_ 5 Theorem 3.1 : R^ 2 (K 3> K y K y K y K y K ) = Proof : According to Theorem 2.2 3 3 R 1,2 (K 3' K 3' K 3' K 3' K 3' V = R 2 (K 3' K 3' V It was shown in [2] that r 3 / 5 p - 5 R 2 (K 3 , K 3 , K p ) = / [6 p > 6 Theorem 3.2 : For p >4,rJ 2 (K 3 , K 3 , K 3 » K 3 , K 4 , K ) = 6. Proof : The coloring scheme in Fig. 3.1 shows that y R lj2 (K 3 , K 3 , K 3 , K 3 , K 4 , K ) > 5 According to Theorem 2.3 R-i o^3' ^3' ^3' ^3' ^4' ^n — R 1^3' ^3' ~ ^ Theorem 3.3 : 3 fs n = p = 4 R, 2 (K 3 , K 3 , K 3 , K», K , K ) =< ''^ J J J q n p [9 n ^ 4, p ^ 5 Proof : That R^ 2 (K 3 , K 3> l< 3 , K 4 , K 4 , K 4 ) = 8 follows from Theorem 4.1, which we shall prove in Section IV. 3 The coloring scheme in Fig. 3.2 shows that R-, 2 ( K 3> K 3> K 3> K 4 K 4 , Kr) > 8. According to Theorem 2.3, R-1 p^3' ^3' ^3' ^4' ^4' ^5' — R 1^3' ^4' = ^' Thus, Ri 9^3' ^3' ^3' ^4' ^4' ^5' ~ Since, for n >_ 4, p > 5 R 1,2 (K 3' K 3' K 3' K 4' K n' K p } - 9 R 1,2 (K 3' K 3' K 3' K 4' K n' V - R 1,2 (K 3' K 3' K 3' K 4' K 4' V we have 3 R 1,2 (K 3' K 3' K 3' K 4' K n' V = 9 ° Theorem 3.4 : R^ 2 (K 3> K 3> K 3 , K & , K R , K ) = 14 for n > 5, p > 5. Proof : It is not difficult to check that the coloring scheme in Fig. 3.3 shows that 3 R-i p(Ko> K~, K-, K r , Kj- , Kr) > 13 (Note that in Fig. 3.3, to simplify the drawings, some of the edges are not shown explicitly. For example, in H , there are edges between v,~ and v ? , v no and v., and v no and v c as indicated but not shown.) 10 4 10 6 According to Theorem 2.3, <2 (K 3' K 3' K 3< K 5' V K p^ ^ R ^ K 3 ^ K 5 } = 14 Theorem 3.5: R? (K , K , K os K , K , K ) = 17 for m > 6, n > 6, l,2 v 333mnp' — — p _> 6. Proof : Accordi ng to Theorem 2.1, for m^6, n ^ 6 , p ^ 6 3 3 R 1,2 (K 3' K 3' K 3' K m' K n' K p ) = R 1 (K 3' K 3' K 3 J = 17 ' (Note that R*(K 3 , Kg) = 6 and R^Kg, K 3 , K 3 ) = 17 [5].) D IV. R 1,2 (I V K j' *V V K 4' V In this section, we determine all Ramsey numbers of the form R ^ 2^ K i' K j' "V K 4' K 4' "V f ° r j 1 4 » J - 4 > £ 1 4 - ( we ignore the degenerate cases in which i = 2, j = 2, or I = 2.) Th eorem 4.1 : R* 2 (K 3 , K-, K 3> K 4 , K 4 , K 4 ) = 8 R, 2^3' ^3' ^4» ^4' ^4» ^4' ~ ^ Proof: The coloring scheme in Fig. 4.1 shows that R-i o(K~, K^, K^, K., K-, K.) > 7 R-j 2^3' ^3' ^4' ^4' ^4' ^4' > ^ We only need to show that R-, 2^3' ^3' ^4' ^4' ^4' ^4' — ^ Consider an arbitrary 3-coloring of the edges of Kg. Let N , N , and N denote the number of a edges, 3 edges, and y edges incident with any one of the vertices, respectively. If N ^4, there is either an a K 3 or a 3 - y K 4 in the coloring. If N >_ 4, there is either a p L or an a - y L in the coloring. If N + N > 6, there is either a B K~ or an a - y K/i in J a y — 3 4 the coloring. If N + N >_ 6, there is either an a K 3 or a B - y K* in the coloring. Consequenctly, we only need to consider the case > (4.1) N a 1 3 N 6 <_ 3 N a + N Y 1 5 N B + N Y 1 5 N a + % + N Y = 7 at every vertex. It follows that N can only assume the values 1, 2, or 3. (1). Suppose N = 3 at one of the vertices. According to the relations in (4.1), N = 2, N = 2 as illustrated in Fig. 4.2(a). The five a p vertices v ? , v~, v., v,., v fi cannot contain a p L nor an a - y K^. Conse- quently, they must contain a 6 circuit of length 5. The two possible cases are illustrated in Fig. 4.2 (b) and (c). For the configuration in Fig. 4.2 (b), we note that if both (v«, v 3 ) and (v-, v,) are y edges, v, , v«» v 3 , v- will form a 6 - y K*. On the other hand, if both (v«, v 3 ) and (v~, v.) are a edges, (Vp, v,-) must be a y edge (so that v«» v 3 , v», v fi will not form an a - 3 K,), and (v«, vj must be a y edge (so that v~, v 3 , v., v 5 will not form an a - 3 K.). However, v 2' v 4' V S' v fi W1 ^ then form a S - y K.. Thus, without loss of generality, we may assume that (v~, v 3 ) is an a edge and (v 3 , v,) is a y edge. It follows that (v., Vr) must be an a edge (otherwise, v 3 , v«, v,-, v g will form a s - y K«). Now, consider the five vertices Vp, v 3 , v,, v 7 , v g . Since they will not contain an a K 3 nor a 3 - y K 3 > they must contain an a circuit of length 5. In view of the fact the colors of the edges (Vp, v~), (v~, v.), (v 3 , v.) have already been fixed, this a circuit of length 5 is uniquely determined to be (vp, v 3 , v^, v 5 , v g ) (up to an interchange of v-, and v g ). It also follows that (v 7 , v g ) must be a y edge. We shows in Fig. 4.3 (a) all the edges whose colors have been determined uniquely, and in Fig. 4.3 (b) those that have not been decided. We note that (Vg, v«) must be an a edge, because otherwise, v 3 , Vr, v g , v g will form a 3 - y K-. It follows that (vp, v g ) must be a y edge so that It is well-known that there is an unique way to color the edges of IC with two colors so that there is no monochromatic triangles, namely, to have a circuit of length 5 of each color. v 2' v 6' v 8 Wl11 not form an a K 3' Conse quently, (v 6 , v 7 ) must be an a edge so that Vp> v 5 , v g , m-j will not form a 3 - y K*. Let us now examine H . We note that (v 5 , v g ) must be a y edge so that Vp, v», v., v g will not form an a - 3 K,. Furthermore, at least one of (Vp, v 7 ) and (Vj-, v 7 ) must be a y edge so that Vp, v., v 5 , v 7 will not form an a - 3 K«. How- ever, we observe in H that Vp, v g , v-,, v g will form an a - y K. (if (v ? , v-J is a y edge), or v 5 , v,-, Vy, v g will form an a - y K. (if (vr, v 7 ) is a y edge). For the configuration in Fig. 4.2 (c), we note first that (Vp, v*) must be an a edge (otherwise, v, , Vp, v 3 , v« will form a p -y L). Again, v 2' v 3' v 4' v 7' v 8 must be contain an a circuit of length 5 and a 3 - y circuit of length 5. The a circuit is uniquely determined to be (vp, v*, v 7 , v 3' v 8^ ^ up t0 an interchange of v 7 and v g ). Also, (v 7 , v g ) must be a y edge We have now the configuration in Fig. 4.4. If (vp, v 7 ) is a 3 edge, Vp, v~, y., v 7 will form an a - 3 K,. If (v*, v«) is a 3 edge, Vp, v 3 , v*, v g will form an a - 3 K-. If both of them are y edges, Vp, v,, v 7 , v g will form an a - y K 4 . (2). Suppose now that no vertex has three y edges incident with it, If N = 1 at a certain vertex, we must have N = N = 3 at that vertex. If N =1 Y a 3 Y at all eight vertices, we must have H as shown in Fig. 4.5 (a). Clearly, there will be an a - 3 K«. If N = 1 at six of the eight vertices, we must have H as shown in Fig. 4.5 (b). Again there will be an a - 3 K,. Suppose that N = 1 at four of the eight vertices. Let v, be one of these vertices. Y 1 We have the configuration in Fig. 4.6 (a). Among Vp, v 3 , v,, v., v^, v 7 , there cannot be an a - 3 K 3 . Therefore, there must be a y Ko. Thus, H can be determined uniquely as that in Fig. 4.6 (b) (up to a permutation of Vp, v 3 , v 4' v 5' v 6' v 7^' How ever, v, , Vp, v,-» v 7 will form an a - 3 K*. We are now left with the case that N = 1 at two of the eight vertices. In this case, Y 10 V. R 1>3 (K 3 , K 3 , K 3 , K 3 , K 4 , K 4 , K p> K q ) We show some simple results in this section to illustrate the case of 4-coloring a complete graph. ,4 Theorem 5. 1 : R-i o(K 3 > K 3 , K 3 , K 3 , K-, K», K«, K. ) - 7 Proof : The coloring scheme in Fig. 5.1 shows that R, 3 (l<3> K 3 , K~, K 3 , K-, K„, K», K«) > 6. Consider any 4-coloring of K-,. Since K-, has 21 edges, one of H , H , H , H has at most 5 edges. We now show that any graph H with 7 vertices and at most 5 edges that contains no triangle must contain an independent set of size 4. We note that if H contains only circuits of even length, the vertices of H can be properly colored with two colors. Consequently, there is an independent set of size 4. If H contains a circuit of length 5, we have the configuration shown in Fig. 5.2. Clearly, v, , v 3 , v fi , v-, form an independent set of size 4. n Theorem 5.2 : R^ 3 (K 3 , K 3 , K 3 , K 3 , K^ K 4 , K 4 , K ) = 8forqi5. 4 / Proof : The coloring scheme in Fig. 5.3 shows that R, 3 (Ko> K 3 , K 3 , K 3 , K 4 , K 4 , K-, Kr) > 7. D To show that R^ 3 (K 3 , K 3 , K 3 , K 3 , K^, K 4> K 4> K ) <^ 8 for q > 5, we prove first the following result: A graph with 8 vertices that contains no triangles nor an independent set of size 4 must have 10 or more edges. Let H be a graph with 8 vertices and 9 or fewer edges. We shall show that if H does not contain any triangle then H must contain an independent set of size 4. We note first that if the vertices of H can be propoerly colored with two colors then H contains an independent set of size 4. We thus assume that H contains either a circuit of length 5 or a circuit of length 7 11 Theorem 4.2 : R^ 2 (K 3> K 4> K 4 , K 4 , K 4 , K 4 ) = 9 Proof : The coloring scheme in Fig. 4.10 shows that R-j 2^3' ^4 ' ^4 ' ^4 ' ^4 ' ^ 4 ) > ^ According to Theorem 2.3 3 2 R-j 2^3* ^4» ^4' ^4» ^4' ^4' — ^1*^3 » ^4' = ^ - Theorem 4.3 : R^ 2 (K 4' K 4' V K 4' K 4> V = 10 Proof : It has been shown in [2] that R?(K 4 , K 4 , K 4 ) = 10. 12 H can be determined uniquely as that shown in Fig. 4.7 (a). There are 3 a edges and 3 B edges incident with Vy. Note that (v, , Vy), (v~, Vy), and (v 3 , Vy) cannot all be a edges since v-, , v 2 , v^, v 7 will form an a - y K* otherwise. Therefore, without loss of generality, we have the coloring shown in Fig. 4.7 (b). We note that (v,, vj and (v^, Vr) must be B edges, and (vg» Vg) and (v 3 , v.) must be a edges.. If (v 3 , vj is an a edge, v 3 , v 4' v 5' v 6 W1 ^ form an a " Y K 4* If ^ v 3' v 5^ 1S a B ed e > v, , v 2 , v^, Vg will form a 6 - y K-. (3). We are now left with the case that N = 2 at all eight vertices •y In this case, H must be that shown in Fig. 4.8 (a). Without loss of generality, we may assume that there are 3 a edges and 2 B edges incident with v fi . We consider the two cases shown in Fin. 4.8 (b) and (c): The case in Fig. 4.8 (b) is identical to the case in Fig. 4.2 (b). For the case in Fig. 4.8 (c), we note first that (v 2 , Vg) must be a B edge. Furthermore, v, , v ? , v 5 , Vy, Vq must contain a B circuit of length 5 which is uniquely (v«, Vg, Vy, v-, , v g ) (up to an interchange of Vy and v g ). We have now the configuration shown in Fig. 4.9. Furthermore, we note that (v«, Vy) and (vg, v g ) must be a edges. If both (v ? , v«) and (v.,, Vg) are 6 edges, v ? , v^, v., Vg will form a B - y K*. Because of symmetry, we can assume that (v ? , v«) is an a edge. So that Vo, v», Vy will not form an a K 3 , (v 4 , Vy 6 K 3 , (v r v 4 a - Y K^, (v-j B K3, (V 3 , Vy * K 3' (v 3' v 8 (v 3 , Vg) must be a B edge. However, v, , v^, v 3 , Vg will then form a B - y must be a B edge. So that v, , v., Vy will not form a must be an a edge. So that v, , v ? , v 3 , v^ will not form an vJ must be a B edge. So that v-, , v,» Vy will not form a must be an a edge. Also, so that v, , v 3 , v g will not form a must be an a edge. So that v 3 , Vg, v g will not form a a K 3 , K 4 . 13 Suppose H contains a circuit of length 5 as shown in Fig. 5.4. Note that since H does not contain any triangle, there are no other edges connecting v, , v~, v 3 , v,, v^. We now examine the three possible cases shown in Fig. 5.5 (a), (b), (c), namely, there is no edge, one edge, and two edges among v g , v-,, v g . For the case in Fig. 5.5 (a), since there are at most four edges connecting the vertices v,-, v^, v g and the vertices v, , v 2' v 3' v 4' V 5* Tnere are two °f v 6' v 7' v 8 t ' iat are ac 'J acent t0 on ^y two of v-., v \j m y Therefore, there is an independent set of size 4. For the case in Fig. 5.5 (b), since there are at most three edges connect- ing the vertices v g , v 7 , Vg and the vertices v, , v 2 , v 3 , v,, v 5 , and since v R can be connected to at most two of the five vertices v, , v^, v~, v., Vj. (otherwise, there is a triangle), one of v g and v 7 together with v R are connected to at most two of the five vertices v, , v^, v 3 , v., v,.. Thus, there is an independent set of size 4. For the case in Fig. 5.5 (c), since there are at most two edges connecting the vertices v fi , v 7 , v R and the vertices v,, v~, v 3 , v.. v 5 , the two vertices v g and v R are connected to at most two of the vertices among v, , v ? , v 3 , v., v g . Consequently, there is an independ- end set of size 4. Suppose H contains a circuit of length 7 as shown in Fig. 5.6. Since the graph in Fig. 5.6 has seven independent sets of size 4, namely, { V V V V { V V V V { V V V V' { V V V V { V V v 7 , Vg}, {v 2 , Vg, v 7 , v g }, {v 3 , v 5 , v 7 , Vg}, the addition of two edges can not destroy all seven of these independent sets. Now, consider any 4-coloring of K Q . Since H D , H , H x cannot all o p Y o have 10 or more edges, there is either a monochromatic triangle or a tri- chromatic K,. n 14 Theorem 5.3 : R* 3 (K 3 , K 3 , K 3 » K 3 , K 4 , K n , K , K ) = 9 for n > 4, p 21 5, q >_ 5 4 Proof : The coloring scheme in Fig. 5.7 shows that R, 3 (K 3 , K 3 , K 3 , K 3 , K 4 , K 4 , K 5 , K 5 ) 8. On the other hand, for n > 4, p > 5, q > 5 R 1,3 (K 3' K 3' K 3' K 3' K 4' K n' V K q ) -W K 4 ) = 9 15 VI. CONCLUDING REMARKS Very little is known about the classical Ramsey numbers R?(K., K., K ) mainly because the value of R-,(K., K., K ) increases very rapidly when i, j, and i increase. The additional constraints in R ? (K., K., K , K , K , K ) which forbid certain bichromatic patterns make the problem more manageable at least for some small cases. It is clear that a great deal more can be done in the direction of determining Ramsey numbers of the form R^ . . , r r 3 J cl 1 , d 2 , .... d h (G 1 , G 2 , ... , 4 G.). For example, it should be possible to determine the value R, 9 (K. , K. , ... , K. ) at least for small i,, i 9 , ... With the exception of the i 2 i 1Q I L results in [3], the case that G, , G^, ... , G. are not complete graphs has been left unexplored. 16 REFERENCES [1] Burr, S. A., "Generalized Ramsey Theory for Graphs--A Survey," in Graphs and Combinatorics , Springer-Verlag, Berlin, 1974, 52-75. [2] Chung, K. M. , and C. L. Liu, "A Generalization of Ramsey Theory for Graphs," J. Discret. Math . , (to appear). [3] Chung, K. M., M. L. Chung, and C. L. Liu, "A Generalization of Ramsey Theory for Graphs—with Complete Graphs and Trees as Forbidden Sub- graphs," The Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing, 1977. [4] Chvatal, V., "On the Ramsey Numbers r(T,K n )," J. of Graph Theory 1, (1), 1977. [5] Greenwood, R. E., and A. M. Gleason, "Combinatorial Relations and Chromatic Graphs," Can. J. Math. 7, 1955, 1-7. r i< v r y 1 1 1 1 1 1 1 1 1 i * 4 4 4 l 1 » < » < » < » \ » i » , » il » 1 » il i 1 i. o o o o o o o 1 all edges connecting vertices in the same column are a edges all edges connecting vertices in the same row are y edges all other edges are B edges Figure 2. I a edges ■- 3 edges all others are Y edges Figure 3.1 > > > >- 4. > oo > CM > > CO > > CnJ > / / / A I \ \ / \ / / > > co. -< / \ \ CM CO a> en ^ / \ - ; > co > > V 4 v 8 V 13 V 7 V 8 V 12 V 3 V ll V 12 V 4 V 10 V 11 ^< 12 \ 3 \ / • ,> V 10 I ? v 13 v 8 v n '** V? ■H V 8 * i > ' i s v 5 v 9 v ll v 6 v ll v 13 v 2 v 9 v 10 v 6 V 10 v 12 V 13f- >v 2 •-4V, 12 '. * -4 v ll V 10 Figure 3.3 a edges 3 edges all others are y edges Figure 4.1 Y .-' Y/ a * V,, Y\ / ; i ,«* * i / \ ! : / e B \i M >4 \ 1 v \3 \ \ 3/ / \\ B 1 / v 6 (c) v 5 Fi gurc 4 .2 V 2< / v 4 \ V6 a edges i $ edges T 7 I i ¥ 2* '» v, • V, (v 2 ,v 6 ) (v 2 ,v 7 ) (v 3 ,v 8 ) (v 5 ,v y ) (V 5' V 8 } (v 6 ,v 7 ) (v 6' v 8 ) a Or y 6 or y 3 or y 3 or y 6 or y Y edges (a) (If a, Vp, v~, Vj will form an a K.J (If a, Vp, Vo, v fi will form an a Ko) (If a, v,, v 5 , v 7 will form an a Ko) (If a, v«, Vj-, v R will form an a Ko) (b) Figure 4.3 y ^ \ / ' n V \ / ■> \ / /y Y \y \ / \ \ ( i » V 3 i /J \\ ) / I \ s ! I B/ \B ; a./ \a ^ * \ V 2«v / < y. \ /%/ \ >s ^a / \ . \ / \ / \ / \ / \ / Figure 4.4 H Y (a) H Y (b) Figure 4.5 • (a) v, • - •• •- v 6 v 7 -• Vr (b) Figure 4.6 v l / \ v 4 v. (a) v 4 (b) Figure 4.7 's 'u • V 5 (b) ■y »-.y * v 7 v 8 Figure 4.9 v 5 v 1# n JV? Y 7 1 /"V v l • V 4 Figure 4.10 Figure 5.1 V 2 v 3 \ \ V 4 V ^ x v 5 V l V 5 v 2 v 4 v l v 4 v 2 v 6 v 3 v e; v 3 v 5 • V- Figure 5.2 \ v l / X y x \ > V 5 / / \ / I 1 T '30 V? •v, /A 4 f Y 6 w II w II w II w== a ^-J v 7 v 2 H 6 Figure 5.3 r i • V 6 •v, Figure 5.4 • V 7 (a) • v, (b) (c) Figure 5.5 Figure 5.6 t V 3 v 2? * v 6 • v, V '1 > V 3 V % v 2 y \. •v ^ >. v l *7% / v 4 • -• >.< ^ ^ V 8 v ^ / V 5 Figure 5.7 BIOGRAPHIC DATA EET 1. Report Ni UIUCDCS-R-77-906 3. Recipient'] \< i < sion Ni \ in,! Sunt ii K A GENERALIZATION OF RAMSEY THEORY FOR GRAPHS 5. Repon Rati December 1977 6. C. L. Liu and K. M. Chung 8. Performing Organization Ri pi, No. ruling Organization Name and Address Department of Computer Science University of Illinois at Urbana-Champaign Urbana, IL 61801 10. Project/ Task Work I 1 1. Contrac t dram No. MCS 73-03408 niK Organization Name and Address National Science Foundation Washington, D.C. 13. Type of Report Hi Period Covered 14. tary Notes Let c, d, , d 2 ,...,d. be integers such that l 1 rv \ Words and Document Analysis. 17a. Descriptors ramsey theory, graph coloring 1 Identifiers Open-Ended Terms ! ' OSA IT Held/Croup V,\- ailafulity Statement i ■'' NTIS-3S I in-7C 19. Security Class (This Report ) UNCLASSIFIED 20. Security (lass (Th Page UNCLASSIFIED 21. No. of Pan 22. Prit USCOMM-DC 40^29-P tm sm Aiij 1 978