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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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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 $kge2$.
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 fan $F_n$ is a graph consisting of $n$ triangles, all having precisely one common vertex. Currently, the best known bounds for the Ramsey number $R(F_n)$ are $9n/2-5 leq R(F_n) leq 11n/2+6$, obtained by Chen, Yu and Zhao. We improve the upper bound to $31n/6+O(1)$.
Let $B_n^{(k)}$ be the book graph which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this paper, we first determine the exact value of $r(B_n^{(2)}, C_m)$ for $frac{8}{9}n+112le mle lceilfrac{3n}{2}rceil+1$ and $n geq 1000$. This answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey numbers, {it Ars Combin.,} {bf 31} (1991), 239--248) in a stronger form when $m$ and $n$ are large. Building upon this exact result, we are able to determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each $k geq 3$. Namely, we prove that for each $k geq 3$, $r(B_n^{(k)}, C_n)= (k+1+o_k(1))n.$ This extends a result due to Rousseau and Sheehan (A class of Ramsey problems involving trees, {it J.~London Math.~Soc.,} {bf 18} (1978), 392--396).
A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In this paper, we study Ramsey numbers of quadrilateral versus books. Previous results give the exact value of $r(C_4,B_n)$ for $1le nle 14$. We aim to show the exact value of $r(C_4,B_n)$ for infinitely many $n$. To achieve this, we first prove that $r(C_4,B_{(m-1)^2+(t-2)})le m^2+t$ for $mge4$ and $0 leq t leq m-1$. This improves upon a result by Faudree, Rousseau and Sheehan (1978) which states that begin{align*} r(C_4,B_n)le g(g(n)), ;;text{where};;g(n)=n+lfloorsqrt{n-1}rfloor+2. end{align*} Combining the new upper bound and constructions of $C_4$-free graphs, we are able to determine the exact value of $r(C_4,B_n)$ for infinitely many $n$. As a special case, we show $r(C_4,B_{q^2-q-2}) = q^2+q-1$ for all prime power $qge4$.
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