  Given a graph H, a graph G is called a Ramsey graph of H if there is a
monochromatic copy of H in every coloring of the edges of G with two colors.
Two graphs G, H are called Ramsey equivalent if they have the same set of
Ramsey graphs. Fox et al. [J. Combin. Theory Ser. B 109 (2014), 120--133] asked
whether there are two non-isomorphic connected graphs that are Ramsey
equivalent. They proved that a clique is not Ramsey equivalent to any other
connected graph. Results of Nesetril et al. showed that any two graphs with
different clique number [Combinatorica 1(2) (1981), 199--202] or different odd
girth [Comment. Math. Univ. Carolin. 20(3) (1979), 565--582] are not Ramsey
equivalent. These are the only structural graph parameters we know that
"distinguish" two graphs in the above sense. This paper provides further
supportive evidence for a negative answer to the question of Fox et al. by
claiming that for wide classes of graphs, chromatic number is a distinguishing
parameter. In addition, it is shown here that all stars and paths and all
connected graphs on at most 5 vertices are not Ramsey equivalent to any other
connected graph. Moreover two connected graphs are not Ramsey equivalent if
they belong to a special class of trees or to classes of graphs with
clique-reduction properties.
