Restricted Size Ramsey Number for Path of Order Three Versus Graph of Order Five

Let $G$ and $H$ be simple graphs. The Ramsey number for a pair of graph $G$ and $H$ is the smallest number $r$ such that any red-blue coloring of edges of $K_r$ contains a red subgraph $G$ or a blue subgraph $H$. The size Ramsey number for a pair of graph $G$ and $H$ is the smallest number $\hat{r}$ such that there exists a graph $F$ with size $\hat{r}$ satisfying the property that any red-blue coloring of edges of $F$ contains a red subgraph $G$ or a blue subgraph $H$. Additionally, if the order of $F$ in the size Ramsey number is $r(G,H)$, then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey number for any pair of small graphs with order at most four. Faudree and Sheehan (1983) continued Harary and Miller\u27s works and summarized the complete results on the (restricted) size Ramsey number for any pair of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for any pair of small forests with order at most five. To continue their works, we investigate the restricted size Ramsey number for a path of order three versus connected graph of order five

Further asymptotic size Ramsey results obtained via linear programming

AbstractRecently, the author (SIAM J. Discrete Math. 16 (2003) 99–113) has asymptotically computed (via linear programming) size Ramsey numbers involving complete bipartite graphs. Here an attempt is made to extend this method to a larger class of problems by considering the ‘simplest’ open case when one of the forbidden graphs is S1,n (the n-star K1,n with an added leaf). Although we obtain new non-trivial results such as, for example, r̂(K2,n,S1,n)=(9+o(1))n and r̂(K3,n,S1,n)=(16+o(1))n, even this ‘simple’ case remains open

Size Ramsey Numbers Involving Double Stars and Brooms

The topics of this thesis lie in graph Ramsey theory. Given two graphs G and H, by the Ramsey theorem, there exist infinitely many graphs F such that if we partition the edges of F into two sets, say Red and Blue, then either the graph induced by the red edges contains G or the graph induced by the blue edges contains H. The minimum order of F is called the Ramsey number and the minimum of the size of F is called the size Ramsey number. They are denoted by r(G, H) and ˆr(G, H), respectively. We will investigate size Ramsey numbers involving double stars and brooms