Almost-Rainbow Edge-Colorings of Some Small Subgraphs

• Autorzy: Elliot Krop , Irina Krop
• Języki publikacji: EN

Abstrakty

EN

Let f(n, p, q) be the minimum number of colors necessary to color the edges of Kn so that every Kp is at least q-colored. We improve current bounds on these nearly “anti-Ramsey” numbers, first studied by Erdös and Gyárfás. We show that [...] , slightly improving the bound of Axenovich. We make small improvements on bounds of Erdös and Gyárfás by showing [...] and for all even n ≢ 1(mod 3), f(n, 4, 5) ≤ n− 1. For a complete bipartite graph G= Kn,n, we show an n-color construction to color the edges of G so that every C4 ⊆ G is colored by at least three colors. This improves the best known upper bound of Axenovich, Füredi, and Mubayi.

Słowa kluczowe

EN

Ramsey theory; generalized Ramsey theory; rainbow-coloring; edge-coloring; Erdös problem

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2013
• Tom: 33
• Numer: 4
• Strony: 771-784

Daty

wydano

2013-09-01

online

2013-10-15

Twórcy

autor

Elliot Krop

• Department of Mathematics, Clayton State University 2000 Clayton State Boulevard, Morrow, GA 30260 USA

autor

Irina Krop

• DePaul University 1 E. Jackson, Chicago, IL 60604 USA

Bibliografia

• [1] M. Axenovich, A generalized Ramsey problem, Discrete Math. 222 (2000) 247-249. doi:10.1016/S0012-365X(00)00052-2[Crossref]
• [2] M. Axenovich, Z. Füredi and D. Mubayi, On generalized Ramsey theory: the bipartite case, J. Combin. Theory (B) 79 (2000) 66-86. doi:10.1006/jctb.1999.1948[Crossref]
• [3] R. Diestel, Graph Theory, Third Edition (Springer-Verlag, Heidelberg, Graduate Texts in Mathematics, Volume 173, 2005).
• [4] P. Erdös, Solved and unsolved problems in combinatorics and combinatorial number theory, Congr. Numer. 32 (1981) 49-62.
• [5] P. Erdös and A. Gyárfás, A variant of the classical Ramsey problem, Combinatorica 17 (1997) 459-467. doi:10.1007/BF01195000[Crossref]
• [6] J. Fox and B. Sudakov, Ramsey-type problem for an almost monochromatic K4, SIAM J. Discrete Math. 23 (2008) 155-162. doi:10.1137/070706628[Crossref][WoS]
• [7] S. Fujita, C. Magnant and K. Ozeki, Rainbow generalizations of Ramsey theory: A survey, Graphs Combin. 26 (2010) 1-30. doi:10.1007/s00373-010-0891-3[Crossref]
• [8] A. Kostochka and D. Mubayi, When is an almost monochromatic K4 guaranteed?, Combin. Probab. Comput. 17 (2008) 823-830. doi:10.1017/S0963548308009413[WoS][Crossref]
• [9] D. Mubayi, Edge-coloring cliques with three colors on all four cliques, Combinatorica 18 (1998) 293-296. doi:10.1007/PL00009822[Crossref]
• [10] R. Wilson, Graph Theory, Fourth Edition (Prentice Hall, Pearson Education Limited, 1996).

Identyfikatory

DOI

10.7151/dmgt.1710