What is Ramsey-equivalent to a clique?

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.

Section: Overview and Observationsmentioning

confidence: 99%

“…Fox et al formulated a question for connected graphs: Question 1. [12] Are there two nonisomorphic connected graphs G and H with G R ∼ H?…”

Section: Introductionmentioning

confidence: 99%

Conditions on Ramsey Nonequivalence

Axenovich

Rollin

Ueckerdt

2017

“…Note H d,d is isomorphic to K d+1 , for which s(K d+1 ) is known to be d 2 [1]. For d = 1, it was recently shown that s(H t,1 ) = t − 1 [6]. For d = 0, it was found s(H t,0 ) = s(K t ) = (t − 1) 2 [11].…”

Section: The Complete Graph With An Added Vertexmentioning

confidence: 99%

“…For a subset S of the vertices and some vertex v ∈ S, define the color pattern c v with respect to S to be the function with domain S that maps a vertex w ∈ S to the color of the edge (v, w). This method was utilized in [6]. For a vertex v ∈ F 1 , consider its color pattern c v with respect to V (R 0 ).…”

Section: The Graph H Td Has Bel Gadgetsmentioning

confidence: 99%

See 1 more Smart Citation

Minimum Degrees of Minimal Ramsey Graphs for Almost‐Cliques

Grinshpun

Raina

Sengupta

2016

Self Cite

For graphs F and H, we say F is Ramsey for H if every 2‐coloring of the edges of F contains a monochromatic copy of H. The graph F is Ramsey H‐minimal if F is Ramsey for H and there is no proper subgraph F′ of F so that F′ is Ramsey for H. Burr et al. defined s(H) to be the minimum degree of F over all Ramsey H‐minimal graphs F. Define Ht,d to be a graph on t+1 vertices consisting of a complete graph on t vertices and one additional vertex of degree d. We show that sfalse(Ht,dfalse)=d2 for all values 1

Chromatic number is Ramsey distinguishing

Savery

2021

A graph G is Ramsey for a graph H if every colouring of the edges of G in two colours contains a monochromatic copy of H. Two graphs H 1 and H 2 are Ramsey equivalent if any graph G is Ramsey for H 1 if and only if it is Ramsey for H 2. A graph parameter s is Ramsey distinguishing if s ( H 1 ) ≠ s ( H 2 ) implies that H 1 and H 2 are not Ramsey equivalent. In this paper we show that the chromatic number is a Ramsey distinguishing parameter. We also extend this to the multicolour case and use a similar idea to find another graph parameter which is Ramsey distinguishing.