On size multipartite Ramsey numbers of {}_{m}K_{1:n}versus P_3 or K_{1:3}
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For given two simple graphs G and H; the size multipartite Ramsey num- ber mj(G;H); is the smallest natural number t such that every 2edge coloring on the edges of the complete balanced multipartite graph Kjt has a monochromatic copy of G in the rst color or H in the second color. Hattingh and Henning (1998) gave the results for the size bipartite Ramsey numbers of stars versus paths, m2(K1;m; Pn); for m; n 2. In 2016, we have derived the size tripartite Ramsey numbers m3(mK1;n; P3); for m 1; n 2; where mK1;n is a disjoint union of m copies of a star K1;n and P3 is a path of order 3. In this paper, we determine the size multipartite Ramsey num- bers mj(mK1;n; P3) and mj(mK1;n;K1;3); for all integers m; n 2 and j 3. We also provide an exact value of m2(mK1;n; P3); for m; n 2 and m3(mK1;n;K1;3); for (m = 1; n 1) or (n = 1;m 1).
Abstract
For given two simple graphs G and H; the size multipartite Ramsey num- ber mj(G;H); is the smallest natural number t such that every 2edge coloring on the edges of the complete balanced multipartite graph Kjt has a monochromatic copy of G in the rst color or H in the second color. Hattingh and Henning (1998) gave the results for the size bipartite Ramsey numbers of stars versus paths, m2(K1;m; Pn); for m; n 2. In 2016, we have derived the size tripartite Ramsey numbers m3(mK1;n; P3); for m 1; n 2; where mK1;n is a disjoint union of m copies of a star K1;n and P3 is a path of order 3. In this paper, we determine the size multipartite Ramsey num- bers mj(mK1;n; P3) and mj(mK1;n;K1;3); for all integers m; n 2 and j 3. We also provide an exact value of m2(mK1;n; P3); for m; n 2 and m3(mK1;n;K1;3); for (m = 1; n 1) or (n = 1;m 1).
참고문헌 (9)
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- G. Chartrand, "Graphs and Digraphs", CRC Press
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- Surahmat, "Star-path size multipartite Ramsey numbers", Applied Math. Sciences (Bulgaria), 8(75), pp.3733~3736 (2014)
- Syafrizal Sy, "The size multipartite Ramsey numbers for paths", Combin. Math. Combin, 55, pp.103~107 (2005)
- Syafrizal Sy, "The size multipartite Ramsey numbers for small paths versus other graphs", Far East J. Appl. Math, 28(1), pp.131~138 (2007)