an:06323195
Zbl 1295.05151
Krop, Elliot; Krop, Irina
Almost-rainbow edge-colorings of some small subgraphs
EN
Discuss. Math., Graph Theory 33, No. 4, 771-784 (2013).
00334323
2013
j
05C55 05C15 05C38
Ramsey theory; generalized Ramsey theory; rainbow-coloring; edge-coloring; Erd??s problem
Summary: Let \(f(n, p, q)\) be the minimum number of colors necessary to color the edges of \(K_n\) so that every \(K_p\) is at least \(q\)-colored. We improve current bounds on these nearly ``anti-Ramsey'' numbers, first studied by Erd??s and Gy??rf??s. We show that \(-3f(n, 5, 9)\geq \frac{7}{4} n\), slightly improving the bound of \textit{M. Axenovich} [Discrete Math. 222, No. 1--3, 247--249 (2000; Zbl 0969.05042)]. We make small improvements on bounds of \textit{P. Erd??s} and \textit{A. Gy??rf??s} [Combinatorica 17, No. 4, 459--467 (1997; Zbl 0910.05034)] by showing \(\frac{5}{6}n+1\leq f(n, 4, 5)\) and for all even \(n \not\equiv 1 \pmod 3\), \(f(n, 4, 5) \leq n- 1\). For a complete bipartite graph \(G= K_{n,n}\), we show an \(n\)-color construction to color the edges of \(G\) so that every \(C_{4} \subseteq G\) is colored by at least three colors. This improves the best known upper bound of \textit{M. Axenovich} et al. [J. Comb. Theory, Ser. B 79, No. 1, 66--86 (2000; Zbl 1023.05101)].
Zbl 0969.05042; Zbl 0910.05034; Zbl 1023.05101