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For a fixed graph $F$ and an integer $t$, the dfn{rainbow saturation number} of $F$, denoted by $sat_t(n,mathfrak{R}(F))$, is defined as the minimum number of edges in a $t$-edge-colored graph on $n$ vertices which does not contain a dfn{rainbow copy} of $F$, i.e., a copy of $F$ all of whose edges receive a different color, but the addition of any missing edge in any color from $[t]$ creates such a rainbow copy. Barrus, Ferrara, Vardenbussche and Wenger prove that $sat_t(n,mathfrak{R}(P_ell))ge n-1$ for $ellge 4$ and $sat_t(n,mathfrak{R}(P_ell))le lceil frac{n}{ell-1} rceil cdot binom{ell-1}{2}$ for $tge binom{ell-1}{2}$, where $P_ell$ is a path with $ell$ edges. In this short note, we improve the upper bounds and show that $sat_t(n,mathfrak{R}(P_ell))le lceil frac{n}{ell} rceil cdot left({{ell-2}choose {2}}+4right)$ for $ellge 5$ and $tge 2ell-5$.
Let $F$ be a fixed graph. The rainbow Turan number of $F$ is defined as the maximum number of edges in a graph on $n$ vertices that has a proper edge-coloring with no rainbow copy of $F$ (where a rainbow copy of $F$ means a copy of $F$ all of whose e
For a graph $H$, a graph $G$ is $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but either removing an edge from $G$ or adding a non-edge to $G$ creates an induced copy of $H$. Depending on the graph $H$, an $H$-induced-saturate
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $delta^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbo
Given graphs $G$ and $H$ and a positive integer $k$, the emph{Gallai-Ramsey number}, denoted by $gr_{k}(G : H)$ is defined to be the minimum integer $n$ such that every coloring of $K_{n}$ using at most $k$ colors will contain either a rainbow copy o
We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum number of colours in an edge-colouring of the complete graph K_n, such that K_n has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph on r vertic