Induced Ramsey Number for a Star Versus a Fixed Graph

We write $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$ for graphs $F, G,$ and $H$, if for any coloring of the edges of $F$ in red and blue, there is either a red induced copy of $H$ or a blue induced copy of $G$. For graphs $G$ and $H$, let $\mathrm{IR}(H,G)$ be the smallest number of vertices in a graph $F$ such that $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$. In this note we consider the case when $G$ is a star on $n$ edges, for large $n$ and $H$ is a fixed graph. We prove that $$ (\chi(H)-1) n \leq \mathrm{IR}(H, K_{1,n}) \leq (\chi(H)-1)^2n + \epsilon n,$$ for any $\epsilon>0$, sufficiently large $n$, and $\chi(H)$ denoting the chromatic number of $H$. The lower bound is asymptotically tight for any fixed bipartite $H$. The upper bound is attained up to a constant factor, for example when $H$ is a clique.

Batas Bawah Bilangan Ramsey pada Graf Bintang S_10 Versus Roda W_12 Lower Bound of the Ramsey Number for the Star Graph S_10 Versus Wheels W_12

Bilangan Ramsey untuk graf terhadap graf , dinotasikan dengan adalah bilangan bulat terkecil sedemikian sehingga untuk setiap graf dengan orde akan memenuhi sifat berikut: memuat graf atau komplemen dari memuat graf .Penelitian ini bertujuan untuk menentukan graf kritis maksimum dan dengan genap. Berdasarkan batas bawah tersebut di tentukan batas atas minimum sehingga diperoleh nilai bilangan Ramsey untuk graf bintang versus , atau . Dengan demikian penentuan batas bawah bilangan Ramsey dilakukan dengan cara batas bawah yang diberikan oleh Chavatal dan Harary, untuk bilangan Ramsey pada graf bintang versus adalah , dengan adalah bilangan kromatik titik graf roda dan adalah kardinalitas komponen terbesar graf . Berdasarkan batas bawah Chavatal dan Harary tersebut dikonstruksi graf kritis untuk dan yang ordenya lebih besar dari nilai batas bawah yang diberikan Chavatal dan Harary. Orde dari graf kritis tersebut merupakan batas bawah terbaik untuk . Kata kunci: Bilangan Ramsey, bintang, roda AbstractRamsey Numbers for a graph to a graph , denoted by is the smallest integer n such that for every graph of order either the following meeet: contains a graph or the complement of contains the graph . This aims of the study to determine the maximum critical graph and . Based on the lower bound of the specified minimum upper bound in order to obtain numerical values for the Ramsey graph Star versus , or . Thus the determination of Ramsey numbers . is done by determine the lower boundary and upper bound. The lower bound given by Chavatal and Harary, for ramsey number for star graph versus wheel is , is a point graph of chromatic number wheel and is the cardinality of the largest component of the graph . Based on the lower bound Chavatal and Harary graph is constructed critical to and are poin greater than the lower bound value given Chavatal and Harary. Order of the critical graph is the best lower bound for . Keywords : Ramsey number, Stars, and Wheels

Star Versus Two Stripes Ramsey Numbers and a Conjecture of Schelp

R. H. Schelp conjectured that if G is a graph with |V(G)| = R(Pn, Pn) such that δ(G) > $$\frac{3|V(G)|}{ 4}$, then in every 2-colouring of the edges of G there is a monochromatic Pn. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree.Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching–matching Ramsey number satisfying R(Sn, nK2, nK2) = 3n − 1. This extends R(nK2, nK2) = 3n − 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma.It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.