Abstract
We show that in every r-coloring of the edges of K n there is a monochromatic double star with at least \(\frac{n(r+1)+r-1}{r^2}\) vertices. This result is sharp in asymptotic for r = 2 and for r≥ 3 improves a bound of Mubayi for the largest monochromatic subgraph of diameter at most three. When r-colorings are replaced by local r-colorings, our bound is \(\frac{n(r+1)+r-1}{r^2+1}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrásfai, B.: Remarks on a paper of Gerencsér and Gyárfás, Ann. Univ. Sci. Eötvös, Budapest 13, 103–107 (1970)
Burr, S.A.: Either a graph or its complement has a spanning broom (manuscript)
Bialostocki, A., Dierker, P., Voxman, W.: Either a graph or its complement is connected: a continuing saga, manuscript (2001)
Bollobás, B., Gyárfás, A.: Highly connected monochromatic subgraphs, Discrete Math. 308, 1722–1725 (2008)
Erdős, P., Faudree, R., Gyárfás, A., Schelp, R.H.: Domination in colored complete graphs, Journal of Graph Theory 13, 713–718 (1989)
Erdős, P., Fowler, T.: Finding large p-colored diameter two subgraphs, Graphs and Combinatorics 15, 21–27 (1999)
Füredi, Z.: Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1, 155–162 (1981)
Grossman, J.W., Harary, F., Klawe, M.: Generalized Ramsey theory for graphs. X. Double stars, Discrete Math. 28, 247–254 (1979)
Gerencsér, L., Gyárfás, A.: On Ramsey type problems, Ann. Univ. Sci. Eötvös, Budapest 10, 167–170 (1967)
Gyárfás, A., Lehel, J., Schelp, R.H., Tuza, Zs.: Ramsey numbers for local colorings, Graphs and Combinatorics 3, 267–277 (1987)
Gyárfás, A., Simonyi, G.: Edge colorings of complete graphs without tricolored triangles, Journal of Graph Theory 46, 211–216 (2004)
Gyárfás, A., Sárközy, G.N.: Size of monochromatic components in local edge colorings, Discrete Math. 308, 2620–2622 (2008)
Gyárfás, A.: Partition coverings and blocking sets in hypergraphs (in Hungarian) Communications of the Computer and Automation Institute of the Hungarian Academy of Sciences 71, 62 pp (1977)
Liu, H., Morris, R., Prince, N.: Highly connected monochromatic subgraphs of multicoloured graphs, submitted
Mubayi, D.: Generalizing the Ramsey problem through diameter, Electronic Journal of Combinatorics 9, R41 (2002)
Author information
Authors and Affiliations
Additional information
András Gyárfás: Research supported in part by OTKA Grant No. K68322.
Gábor N. Sárközy: Research supported in part by the National Science Foundation under Grant No. DMS-0456401 and by OTKA Grant No. K68322.
Rights and permissions
About this article
Cite this article
Gyárfás, A., Sárközy, G.N. Size of Monochromatic Double Stars in Edge Colorings. Graphs and Combinatorics 24, 531–536 (2008). https://doi.org/10.1007/s00373-008-0811-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-008-0811-y