For a positive integer n and graph B, fa (n) is the least integer m such that any graph of order n and minimal degree m has a copy of B. It will be show that if B is a bipartite graph with parts of order k and i (k-1), then there exists a positive constant c, such that for any tree T of order n and for any j (0-j _-(k-1)), the Ramsey number r(T, B) < n + c-(fe (n)) 't(k-1) if A(T)-(n/(k-j-1))-(...

The well-known Chvátal-Erdős theorem states that if the stability number α of a graph G is not greater than its connectivity then G is hamiltonian. In 1974 Erdős showed that if, additionally, the order of the graph is sufficiently large with respect to α, then G is pancyclic. His proof is based on the properties of cycle-complete graph Ramsey numbers. In this paper we show that a similar result...

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp in [2], where it is shown that cycles (except for C4) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle Cn, u...

We consider combinatorial avoidance and achievement games based on graph Ramsey theory: The players take turns in coloring still uncolored edges of a graph G, each player being assigned a distinct color, choosing one edge per move. In avoidance games, completing a monochromatic subgraph isomorphic to another graph A leads to immediate defeat or is forbidden and the first player that cannot move...

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some conjectures and results concerning both Ramsey saturated and unsaturated graphs. In particular, we show that cycles Cn and paths Pn on n vertices are Ramsey unsaturated for all n ≥ 5. 1 Results and Conjectures Throughout this article, r...

The Ramsey number r(H) of a graph H is the minimum positive integer N such that every twocoloring of the edges of the complete graph KN on N vertices contains a monochromatic copy of H . A graph H is d-degenerate if every subgraph of H has minimum degree at most d. Burr and Erdős in 1975 conjectured that for each positive integer d there is a constant cd such that r(H) ≤ cdn for every d-degener...

One of the earliest and best-known application of the probabilistic method is the proof of existence of a 2 logn-Ramsey graph, i.e., a graph with n nodes that contains no clique or independent set of size 2 logn. The explicit construction of such a graph is a major open problem. We show that a reasonable hardness assumption implies that in polynomial time one can construct a list containing pol...