Conditions on Ramsey Nonequivalence
نویسندگان
چکیده
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 nonisomorphic connected graphs that are Ramsey equivalent. They proved that a clique is not Ramsey equivalent to any other connected graph. Results of Nešetřil 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 two connected graphs are not Ramsey equivalent if they have at most 5 vertices, if they belong to a special class of trees, or to classes of graphs with cliquereduction properties. An infinite class of graphs is given such that any graph in this class is not Ramsey equivalent to any other connected graph.
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ورودعنوان ژورنال:
- Journal of Graph Theory
دوره 86 شماره
صفحات -
تاریخ انتشار 2017