Conditions on Ramsey Nonequivalence

Abstract

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. [565][566][567][568][569][570][571][572][573][574][575][576][577][578][579][580][581][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.

On minimal Ramsey graphs and Ramsey equivalence in multiple colours

“…For all 1 i < j q, join e i and e j by a negative signal sender S − = S − (q + 1, K k , k). And for all 1 i q and every edge e ∈ F (1) i…”

“…By inductive hypothesis of G q , there exists a (K k • K 2 )-free colouring c 0 : E( G) → [q] of the edges in V 0 . For all 1 i q, colour the edges of F (1) i ∪ • • • ∪ F (k−2) i and the edge e i in colour i. Colour all edges between any V i and V j , 0 i < j k − 2, with colour q + 1. Note that all pairs of edges that are joined by copies of S + have the same colour.…”

“…After permuting colours we may henceforth assume that the edge e i has colour i for 1 i q. Furthermore, c is K k -free on every copy of S + which joins e and e i , for each i ∈ [q] and e ∈ F (1) i 2) i . This implies that the graph…”

Chromatic number is Ramsey distinguishing

“…If a graph has no component of size at least 3, then clearly it can only be Ramsey equivalent to graphs which also have this property. Any forest with a component of size at least 3 has 2‐density equal to 1, and it was shown by Axenovich, Rollin and Ueckerdt in [1] that no forest is Ramsey equivalent to a graph containing a cycle. So in fact the 2‐density is a Ramsey distinguishing parameter among graphs which have a component of size at least 3, and graphs which do not have such a component are not Ramsey equivalent to any which do.…”

“…The Ramsey distinguishing properties of the chromatic number were investigated by Axenovich, Rollin and Ueckerdt in [1]. They observe that if and are graphs with bipartite and then since any sufficiently large complete bipartite graph is Ramsey for [2] but does not contain , and hence chromatic number is distinguishing for bipartite graphs.…”