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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 of $G$ or a monochromatic copy of $H$. We consider this question in the cases where $G in {P_{4}, P_{5}}$. In the case where $G = P_{4}$, we completely solve the Gallai-Ramsey question by reducing to the $2$-color Ramsey numbers. In the case where $G = P_{5}$, we conjecture that the problem reduces to the $3$-color Ramsey numbers and provide several results in support of this conjecture.
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Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we consider $gr_k(K_3 : H)$ where $H$ is a connected graph with five vertices and at most six edges. There are in total thirteen graphs in this graph class, and the Gallai-Ramsey numbers for some of them have been studied step by step in several papers. We determine all the Gallai-Ramsey numbers for the remaining graphs, and we also obtain some related results for a class of unicyclic graphs.
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.
We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)leq R_k(P_n)leq R_k(C_n)leq kn+o(n)$. The upper bound was recently improved by Sarkozy who showed that $R_k(C_n)leqleft(k-frac{k}{16k^3+1}right)n+o(n)$. Here we show $R_k(C_n) leq (k-frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant.
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.
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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