Large rainbow matchings in large graphs

A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough (namely, $n\geq 4.25k^2$), then each $n$-vertex graph $G$ with minimum color degree at least $k$ contains a rainbow matching of size at least $k$

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Long rainbow cycles in proper edge-colorings of complete graphs

We show that any properly edge-colored Kn contains a rainbow cycle with at least (4=7 − o(1))n edges. This improves the lower bound of n=2 − 1 proved in [1]

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Complexity of Computing the Anti-Ramsey Numbers for Paths

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erd\" os, Simonovits and S\' os. For given graphs $G$ and $H$ the \emph{anti-Ramsey number} $\textrm{ar}(G,H)$ is defined to be the maximum number $k$ such that there exists an assignment of $k$ colors to the edges of $G$ in which every copy of $H$ in $G$ has at least two edges with the same color. There are works on the computational complexity of the problem when $H$ is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number $\textrm{ar}(G,P_k)$, where $P_k$ is a path of length $k$. First, we observe that when $k = \Omega(n)$, the problem is hard; hence, the challenging part is the computational complexity of the problem when $k$ is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing $\textrm{ar}(G,P_k)$ for every integer $k>2$ is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating $\textrm{ar}(G,P_3)$ to a factor of $n^{-1/2 - \epsilon}$ is hard already in $3$-partite graphs, unless P=NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant $k$. Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. We introduce the notion of color connected coloring and employing this structural property. We obtain a linear time algorithm to compute $\textrm{ar}(G,P_k)$, for every integer $k$, when the host graph, $G$, is a tree