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A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given a graph $H$ and an integer $kgeq 1$, the Gallai-Ramsey number $GR_k(H)$ of $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $W_{2n} $ denote an even wheel on $2n+1ge5$ vertices. In this note, we study Gallai-Ramsey number of $W_{2n}$ and completely determine the exact value of $GR_k(W_4)$ for all $kge2$.
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For a graph $H$ and an integer $kge1$, the $k$-color Ramsey number $R_k(H)$ is the least integer $N$ such that every $k$-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $mge4$ vertices and let $Theta_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. Unlike Ramsey number of odd cycles, little is known about the general behavior of $R_k(C_{2n})$ except that $R_k(C_{2n})ge (n-1)k+n+k-1$ for all $kge2$ and $nge2$. In this paper, we study Ramsey number of even cycles with chords under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. For an integer $kgeq 1$, the Gallai-Ramsey number $GR_k(H)$ of a graph $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. We prove that $GR_k(Theta_{2n})=(n-1)k+n+1$ for all $kgeq 2$ and $ngeq 3$. This implies that $GR_k(C_{2n})=(n-1)k+n+1$ all $kgeq 2$ and $ngeq 3$. Our result yields a unified proof for the Gallai-Ramsey number of all even cycles on at least four vertices.
Given a graph $G$ and a positive integer $k$, the emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for double stars $S(n,m)$, where $S(n,m)$ is the graph obtained from the union of two stars $K_{1,n}$ and $K_{1,m}$ by adding an edge between their centers. We also provide the sharp result in some cases.
The Turan number of a graph $H$, $text{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. A wheel $W_n$ is an $n$-vertex graph formed by connecting a single vertex to all vertices of a cycle $C_{n-1}$. Let $mW_{2k+1}$ denote the $m$ vertex-disjoint copies of $W_{2k+1}$. For sufficiently large $n$, we determine the Turan number and all extremal graphs for $mW_{2k+1}$. We also provide the Turan number and all extremal graphs for $W^{h}:=bigcuplimits^m_{i=1}W_{k_i}$ when $n$ is sufficiently large, where the number of even wheels is $h$ and $h>0$.
A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses at most $k$ colors. For an integer $kgeq 1$, the Gallai-Ramsey number $GR_k(H)$ of a given graph $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $mge4$ vertices and let $Theta_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. We prove that $GR_k(Theta_{2n+1})=ncdot 2^k+1$ for all $kgeq 1$ and $ngeq 3$. This implies that $GR_k(C_{2n+1})=ncdot 2^k+1$ all $kgeq 1$ and $ngeq 3$. Our result yields a unified proof for the Gallai-Ramsey number of all odd cycles on at least five vertices.
We find the exact value of the Ramsey number $R(C_{2ell},K_{1,n})$, when $ell$ and $n=O(ell^{10/9})$ are large. Our result is closely related to the behaviour of Turan number $ex(N, C_{2ell})$ for an even cycle whose length grows quickly with $N$.
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