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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.
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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 Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices.
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 $kge2$.
Given a positive integer $ r $, the $ r $-color size-Ramsey number of a graph $ H $, denoted by $ hat{R}(H, r) $, is the smallest integer $ m $ for which there exists a graph $ G $ with $ m $ edges such that, in any edge coloring of $ G $ with $ r $ colors, $G$ contains a monochromatic copy of $ H $. Haxell, Kohayakawa and L uczak showed that the size-Ramsey number of a cycle $ C_n $ is linear in $ n $ i.e. $ hat{R}(C_n, r) leq c_rn $, for some constant $ c_r $. Their proof, however, is based on the Szemeredis regularity lemma and so no specific constant $ c_r $ is known. Javadi, Khoeini, Omidi and Pokrovskiy gave an alternative proof for this result which avoids using of the regularity lemma. Indeed, they proved that if $ n $ is even, then $ c_r $ is exponential in $ r $ and if $ n $ is odd, then $ c_r $ is doubly exponential in $ r $. oindent In this paper, we improve the bound $c_r$ and prove that $c_r$ is polynomial in $r$ when $n$ is even and is exponential in $r$ when $n$ is odd. We also prove that in the latter case, it cannot be improved to a polynomial bound in $r$. More precisely, we prove that there are some positive constants $c_1,c_2$ such that for every even integer $n$, we have $c_1r^2nleq hat{R}(C_n,r)leq c_2r^{120}(log^2 r)n$ and for every odd integer $n$, we have $c_1 2^{r}n leq hat{R}(C_n, r)leq c_22^{16 r^2+2log r}n $.
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