UNIVERSITY OF, ILLINOIS LIBRARY At URBANA-CHAMPAIGN The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below.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 Digitized by the Internet Archive in 2013 http://archive.org/details/generalizationof864chun 5 /v< c f ftl^ L% Ca^ UIUCDCS-R-77-864 A GENERALIZATION OF RAMSEY THEORY FOR GRAPHS— WITH STARS AND COMPLETE GRAPHS AS FORBIDDEN SUBGRAPHS UILU-ENG 77 1718 by K. M. Chung, M. L. Chung, and C. L. Liu May 1977 Jbrary JUN 2 DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS A GENERALIZATION OF RAMSEY THEORY FOR GRAPHS--WITH STARS AND COMPLETE GRAPHS AS FORBIDDEN SUBGRAPHS by K. M. Chung, M. L. Chung, and C. L. Liu Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 61801 May 1977 * This work was supported in part by the National Science Foundation under Grant Number MCS-73-03408. I. INTRODUCTION In [1], the following generalization of Ramsey Theory for graphs was introduced: Let c and d be two integers such that c > d >^ 1 . Let t denote the binomial coefficient l^j . Given c distinct colors, we order the t subsets of d colors in some arbitrary manner. Let G, , G ? , ... , G. be graphs. The d-chromatic Ramsey number, denote by r ,, (G, , G ? , ... , G.), is defined to be the least number p such that if the edges of the complete graph K are colored in any fashion with c colors; then for some i the sub- graph whose edges are colored with i subset of colors contains a G. . In [1], the case c = 3, d = 2, G, , G ? , G~ being complete graphs was studied. In this paper we further study cases where G, , G 2 , G~ are stars and complete 3 graphs. Thus, for instance, r^K, ., K-, ., K, ) denotes the smallest p such that if the edges of the complete graph K are colored with the colors a, 6, and y. there is either an a - 6 K-, . or an a - y K, . or a 6 - y K in the graph K . •Xr K u K. ) - 1 » «.' 1 + J rio+j+ifl 4 1 i + J; i,j both even £ > 1 + j; i,j not both even * < 1 + J; d,j,a) c s * < i + j; d,j,t) 4 s where S - {(lj,*)| T + j + e , 3 mod 4 and not all i, j, g are odd Proof: For the cases l > i + j , we have [2]: r(R i.i- ic i.j , "{;;j'' if i ,j are both even otherwise Therefore by Theorem 2.2: 3, fi + j - 1 if e > i + j - 1, and i , j are both even 2 l»i' l.j' l,r I 1 ' + J if ^ 1 i + j, i,j are not both even For cases i < i + j, we establish lower bounds by giving a critical coloring for each of case when (k,j,0 E S and (i,j,0 I S. 1/. For the case of i + j + i = 3 mod 4, we consider the colorinq of K , P P = 2^ 1+ J +£ " 3 )- Since p is even, according to Lemma 3.2, we can decompose K into regular graphs G o§ G p , and G , whose degrees are ^-(i+j-A-3), i(i-j+£-l), and -(-i+j+£-i), respectively. We color G with color a, G D with color 6, and G with color y. It is easy to check that N + N < i, N + N < i and a 3 a Y N a + N < i at each vertex. p Y For the case of (i,j,0 e S, we consider the coloring of K , 1 P P = F 2" C "" "*" J + a )~I - 1. There are 4 subcases: Casejj If i + j + £ e mod 4, p = ^(i+j+a) - 1 and p is odd. Let a = 2"(i + J-0 - 1, b = 2-(i-j+0 - 1, c = g-H+j+Jl) - 1. Since a + b + c = l(i+j+Jl) - 3 which is odd, either one or all of a, b, and c are odd. There are 2 possibilities. Suppose a is odd, but b and c are even. We decompose K into regular qraphs G , G„, G of even degrees N = a + 1, N = b, and N = c. We color G with color a, G with color 3, and G with color y. Similar coloring can be made if either b or c is odd. Suppose a, b, and c are all odd. We first consider the coloring of a complete subgraph K , of K . We decompose K -, into regular graph G , G and G of degree a, b, and c respectively. G is then colored with color a, G with 3, and G with y. We color the p - 1 edges that connect the remain- ing vertex to the other vertices in the following fashion: a + 1 edges are colored with a, b edges with 3, and c edges with r. Since at each vertex of K , either N = a + 1, N = b, N = c, or N = a, N = b + 1, N = c, or p a 3 Y a 3 Y N = a, N = b, N = c + 1, it can be easily checked that there is indeed no a 3 Y a - 3 K, ., nor an a - y K, . , nor a 3 - y K, . I j 1 I , J I , x. Case 2 : Suppose i+j+£=lmod4,p= ["^(i+J+Ol = i(i + J +jl " 1 ) and is even. We decompose K into regular graphs G , G , and G whose degrees are «-(i+j-£-l ), oO-j+fc-l) and 2"(-i + J + £-l ) ancl which are then colored with colors a, 3, and Y, respectively. Case 3 : If i + j + I = 2 mod 4, p = p-(i+j+£) - 1 and is even. We decompose K into regular graphs G , G , and G whose degrees are ~-(i+j-0> -o(i-J + £-l) and jK-i+j+Jl-1) and which are then colored with colors a, 3, and y ^(-i+j+a-l ) > respectively. Case 4 : If i + j + i ~ 3 mod 4 and i,j>£ all odd, p = 2"( 1+ J +£ ) " 1 and is odd. We decompose K into regular graphs G , G , and G , whose degrees are ^(i+j-Ji-l ) , ^(i-j+n-l ) > and ^( -i +j+ii-l ), all even, and which are then colored with colors a, 3, and y» respectively. We next show that for (i,j,£) e S, it is not possible to color a K , p = Pp^ ( i + J + ^)~l - 1 that does not contain an a - 3 K, . , or an a - y K, ., or a B - y K, „. From (3.1), (3.2). and (3.3), since N + N + N = p - 1 = 1,2, a 8 y i(i+j+£+l) - 2, we must have N > tf-i+j+H-l), N > ki-j+A-1) and N > i(i+j-a-l) s at each vertex. But since N + N + N = 2-( 1 '+J +£+1 ) _ 2 > WG must have at each vertex, N = l(+i+j-A-l) s N n - i(i-j+J£-l ) , N = o(-' i+ J +,i -l ) • If i = 2n, + 1, j = 2n ? , t = 2n 3 then i + j + I = 3 mod 4 implies (n-j+n^+nj is odd. Therefore N is also odd. Since p is odd, then there is no coloring for G . We still have to consider the case when i = i + j and both i and j are even. Consider coloring a K , p = i + j - 1. Applying the above argument, we must have at each vertex N = 0, N = 9-(i-j+£.) - 1 = i - 1, and N = ~- o. p c y c (-i+j+Jt) - 1 = j - 1. But again since p and N are both odd, no coloring is possible. □ I V . RAMSEY NUMBERS FOR STARS AND COMPLETE GRAPHS Results have been obtained for evaluating exact values of the 3 Ramsey numbers r (K, . , K., K.), for small i and j. Since the proofs c 1,1 J I are lengthy and involved, they will not be included. See [3] for details Without lost of generality, we assume that j < £. The results are: Theorem 4.1 r 2 (K l,2' K j' K £ } r 2j - 1 £ > 2j - 1 2, r 2 (f V K j' K * } = \ I (j+£_1) A < 2J - 1 and j + £ 2|"J^*] - 1 otherwise = 1 (mod 3) Theorem 4.2; r 2 (K i,3- V V ■ 2 J Theorem 4.3: '= 2i + 1 = 2i r 2 3 (K 1}i , K 3 , K £ )^ =21 - 1 1 % + 1 - 2 < £ + i - 1 £ >_ 2i +1 i = 2i i < £ < 2i - 1 £ < i , i even , £ odd otherwise Corollary 4.3.1 r r 2 (K l,4' K 3' K £ ) £ = 3 4 £ £ <_ 7 £ = 8 £ > 9 REFERENCE [1] K. H. Chung and C. L. Liu, A Generalization of Ramsey Number For Graphs, to appear. [2] F. Harrary, Recent Results On Generalized Ramsey Theory Graphs, Graph Theory and Applications (Y. Alavi, D. R. Lick, and A. T. White, eds."), Springer-Verlag, Berlin, 1972. [3] M. L. Chung, A Generalization Of Ramsey Number For Graphs, Master Thesis , Department of Computer Science, University of Illinois at Urbana-Champaign, to appear. BLIOGRAPHIC DATA :eet "Title jnd Subtitle 1. Report No. UIUCDCS-R-77-864 3. Recipient's Accession No. 5- Report Date A Generalization of Ramsey Theory For Graphs—With Stars and Complete Graphs as Forbidden Subgraphs May 1977 Author(s) K. M. Chung, M. L. Chung, and C. L. Liu 8. Performing Organization Rept. No. • Performing Organization Name and Address Department of Computer Science University of Illinois at Urbana-Champaign Urbana, IL 61801 10. Project/Task/Work Unit No. 11. Contract/Grant No. MCS-73-03408 Sponsoring Organization Name and Address National Science Foundation Washington, DC 13*. Type of Report & Period Covered 14. . Supplementary Notes Abstracts Let c and d be two integers such that c >_ d j> 1 . Let t denote the binomial coefficient ^J. Given c distinct colors, we order the t subsets of d colors in some arbitrary manner. Let G ] , G 2 , ..., G t be graphs. The d-chromatic Ramsey number , denoted by r^(G ] , G 2 , ..., G t ) is defined to be the least number p such that if the edges of complete graph K are colored in any fashion with c colors, then for some i the subgraph whose edges are colored with the i th subset of colors contains a G i . In this report, we study the case c = 3, d = 2. Results are obtained for cases where G ] , G 2 , G 3 are complete graphs or stars. '. Key Words and Document Analysis. 17a. Descriptors Ramsey numbers and Critical coloring, 'b. Identifiers/Open-Ended Terms "c COSATI Field/Group 1- Availability Statement 5RM NTIS-15 ( 10-70) 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 22. Price USCOMM-DC 40329-P7I J l/l >>*?>