In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs. Revision History • Revision 3: March, 2015. • Revision 2: October, 2014. • Revision 1: July, 2011. • Original: Graphs and Combinatorics, January, 2010. [73] If you have corrections, updates or new results which fit the scope of this work, please contact Colton Magnant at cmagnant@georgiasouthern.edu.

/pdf/rainbow-generalizations-of-ramsey-theory-a-dynamic-survey-gndmlox4ie.pdf

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

Let $$\mathrm{pr}(K_{n}, G)$$
 be the maximum number of colors in an edge-coloring of $$K_{n}$$
 with no properly colored copy of G. For a family $${\mathcal {F}}$$
 of graphs, let $$\mathrm{ex}(n, {\mathcal {F}})$$
 be the maximum number of edges in a graph G on n vertices which does not contain any graphs in $${\mathcal {F}}$$
 as subgraphs. In this paper, we show that $$\mathrm{pr}(K_{n}, G)-\mathrm{ex}(n, \mathcal {G'})=o(n^{2}), $$
 where $$\mathcal {G'}=\{G-M: M \text { is a matching of }G\}$$
 . Furthermore, we determine the value of $$\mathrm{pr}(K_{n}, P_{l})$$
 for sufficiently large n and the exact value of $$\mathrm{pr}(K_{n}, G)$$
 , where G is $$C_{5}, C_{6}$$
 and $$K_{4}^{-}$$
 , respectively. Also, we give an upper bound and a lower bound of $$\mathrm{pr}(K_{n}, K_{2,3})$$
 .

/pdf/on-the-maximal-colorings-of-complete-graphs-without-some-qqtbzk3qf8.pdf

On the Maximal Colorings of Complete Graphs Without Some Small Properly Colored Subgraphs

Let $pr(K_{n}, G)$ be the maximum number of colors in an edge-coloring of $K_{n}$ with no properly colored copy of $G$. In this paper, we show that $pr(K_{n}, G)-ex(n, \mathcal{G'})=o(n^{2}), $ where $\mathcal{G'}=\{G-M: M \text{ is a matching of }G\}$. Furthermore, we determine the value of $pr(K_{n}, P_{l})$ for $l\ge 27$ and $n\ge 2l^{3}$ and the exact value of $pr(K_{n}, G)$, where $G$ is $C_{5}, C_{6}$ and $K_{4}^{-}$, respectively. Also, we give an upper bound and a lower bound of $pr(K_{n}, K_{2,3})$.

Minimal colorings for properly colored subgraphs in complete graphs

Given two graphs $G$ and $H$ with $H\subseteq G$ we consider the anti-Ramsey function $AR(G,H)$ which is the maximum number of colors in any edge-coloring of $G$ so that every copy of $H$ receives the same color on at least one pair of edges. The classical Turan function for a graph $G$ and family of graphs $\mathcal{F}$, written $ex(G,\mathcal{F})$, is defined as the maximum number of edges of a subgraph of $G$ not containing any member of $\mathcal{F}$. We show that there exists a constant $c>0$ so that $AR(K_n,K_{s,t})-ex(K_n,K_{s,t})<cn$ and $c$ depends only on $s$ and $t$, which implies $AR(K_n,K_{s,t})\leq cn^{2-\frac{1}{s}}$, for $s\leq t$ by a result of K\H ovari, Sos, and Turan.

On anti-Ramsey numbers for complete bipartite graphs and the Turan function

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On anti-Ramsey numbers for complete bipartite graphs and the Turan function