The minimum degree of Ramsey-minimal graphs (original) (raw)
Related papers
Journal of the Brazilian …, 2001
As usual, for graphs Γ, G, and H, we write Γ → (G, H) to mean that any red-blue colouring of the edges of Γ contains a red copy of G or a blue copy of H. A pair of graphs (G, H) is said to be Ramsey-infinite if there are infinitely many minimal graphs Γ for which we have Γ → (G, H). Let ≥ 4 be an integer. We show that if H is a 2connected graph that does not contain induced cycles of length at least , then the pair (C k , H) is Ramseyinfinite for any k ≥ , where C k denotes the cycle of length k.
On the minimum degree of minimal Ramsey graphs for multiple colours
Journal of Combinatorial Theory, Series B, 2016
A graph G is r-Ramsey for a graph H, denoted by G → (H) r , if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramseyminimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s r (H) denote the smallest minimum degree of G over all graphs G that are r-Ramseyminimal for H. The study of the parameter s 2 was initiated by Burr, Erdős, and Lovász in 1976 when they showed that for the clique s 2 (K k) = (k − 1) 2. In this paper, we study the dependency of s r (K k) on r and show that, under the condition that k is constant, s r (K k) = r 2 • polylog r. We also give an upper bound on s r (K k) which is polynomial in both r and k, and we determine s r (K 3) up to a factor of log r.
Minimal vertex Ramsey graphs and minimal forbidden subgraphs
Discrete Mathematics, 2004
Let P be a property of graphs. A graph G is vertex (P; k)-colourable if the vertex set V (G) of G can be partitioned into k sets V1; V2; : : : ; V k such that the subgraph G[Vi] of G belongs to P, i = 1; 2; : : : ; k. If P is a hereditary property, then the set of minimal forbidden subgraphs of P is deÿned as follows: F(P) = {G: G ∈ P but each proper subgraph H of G belongs to P}. In this paper we investigate the property On: each component of G has at most n+1 vertices. We construct minimal forbidden subgraphs for the property (O k n) "to be (On; k)-colourable". We write G v → (H) k , k ¿ 2, if for each k-colouring V1; V2; : : : ; V k of a graph G there exists i, 1 6 i 6 k, such that the graph induced by the set Vi contains H as a subgraph. A graph G is called (H) k-vertex Ramsey minimal if G v → (H) k , but G v 9 (H) k for any proper subgraph G of G. The class of (P3) k-vertex Ramsey minimal graphs is investigated.
On Ramsey-Minimal Infinite Graphs
The Electronic Journal of Combinatorics
For fixed finite graphs GGG, HHH, a common problem in Ramsey theory is to study graphs FFF such that Fto(G,H)F \to (G,H)Fto(G,H), i.e. every red-blue coloring of the edges of FFF produces either a red GGG or a blue HHH. We generalize this study to infinite graphs GGG, HHH; in particular, we want to determine if there is a minimal such FFF. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair (G,H)(G,H)(G,H) to have a Ramsey-minimal graph. We use these to prove, for example, that if G=SinftyG=S_\inftyG=Sinfty is an infinite star and H=nK2H=nK_2H=nK2, ngeqslant1n \geqslant 1ngeqslant1 is a matching, then the pair (Sinfty,nK2)(S_\infty,nK_2)(S_infty,nK_2) admits no Ramsey-minimal graphs.
Ramsey minimal graphs for 2K_2 versus 2C_n
Applied Mathematical Sciences, 2015
Let G and H be two given graphs. The notation F → (G, H) means that any red-blue coloring on the edges of F will create either a red subgraph G or a blue subgraph H in F. A graph F is a Ramsey (G, H)minimal graph if F → (G, H) and F * → (G, H) for any proper subgraph F * ⊂ F. The set of all (G, H)-minimal graphs is denoted by R(G, H). In this paper we give some necessary conditions for the members of R(2K 2 , 2C n) for n ≥ 3 and determine some graphs in R(2K 2 , 2C 3) and R(2K 2 , 2C 4).
Vertex Folkman Numbers and the Minimum Degree of Minimal Ramsey Graphs
SIAM Journal on Discrete Mathematics, 2018
We investigate the smallest possible minimum degree of r-color minimal Ramsey-graphs for the k-clique. In particular, we obtain a bound of the form O(k 2 log 2 k , which is tight up to a (log 2 k)-factor whenever the number r ≥ 2 of colors is fixed. This extends the work of Burr, Erdős, and Lovász who determined this extremal value for two colors and any clique size, and complements that of Fox, Grinshpun, Liebenau, Person, and Szabó who gave essentially tight bounds when the order k of the clique is fixed. As a side product our result also yields an improved upper bound on the vertex Folkman number F (r, k, k + 1) of the k-clique. The proof relies on a reformulation of the corresponding extremal function by Fox et al, and combines and refines methods used by Dudek, Eaton, and Rödl.
Ramsey Properties of Families of Graphs
Journal of Combinatorial Theory, Series B, 2002
For a graph F and natural numbers a 1 ; . . . ; a r ; let F ! ða 1 ; . . . ; a r Þ denote the property that for each coloring of the edges of F with r colors, there exists i such that some copy of the complete graph K ai is colored with the ith color. Furthermore, we write ða 1 ; . . . ; a r Þ ! ðb 1 ; . . . ; b s Þ if for every F for which F ! ða 1 ; . . . ; a r Þ we have also F ! ðb 1 ; . . . ; b s Þ: In this note, we show that a trivial sufficient condition for the relation ða 1 ; . . . ; a r Þ ! ðb 1 ; . . . ; b s Þ is necessary as well. # 2002 Elsevier Science (USA) # 2002 Elsevier Science (USA)
Discrete Mathematics, 2016
In this paper, we study an analogue of size-Ramsey numbers for vertex colorings. For a given number of colors r and a graph G the vertex size-Ramsey number of G, denoted byR v (G, r), is the least number of edges in a graph H with the property that any r-coloring of the vertices of H yields a monochromatic copy of G. We observe that Ω r (∆n) =R v (G, r) = O r (n 2) for any G of order n and maximum degree ∆, and prove that for some graphs these bounds are tight. On the other hand, we show that even 3-regular graphs can have nonlinear vertex size-Ramsey numbers. Finally, we prove thatR v (T, r) = O r (∆ 2 n) for any tree of order n and maximum degree ∆, which is only off by a factor of ∆ from the best possible.