A note on Ramsey size‐linear graphs

We show that if G is a Ramsey size‐linear graph and x,y ∈ V (G) then if we add a sufficiently long path between x and y we obtain a new Ramsey size‐linear graph. As a consequence we show that if G is any graph such that every cycle in G contains at least four consecutive vertices of degree 2 then G is Ramsey size‐linear. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 1–5, 2002     

Ramsey graphs and block designs

Symmetric (ν, κ, λ)-block designs admitting polarity maps are shown to be closely related to certain Ramsey numbers for bipartite graphs. In particular, if there exists a (ν, κ, λ)-difference set in an abelian group of order ν, then the Ramsey number R(K_2,λ+1, K_1,ν?k+1) is either 1 + ν or 2 + ν.     

Ramsey unsaturated and saturated graphs

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some conjectures and results concerning both Ramsey saturated and unsaturated graphs. In particular, we show that cycles C_n and paths P_n on n vertices are Ramsey unsaturated for all n ≥ 5. © 2005 Wiley Periodicals, Inc.     

New families of graphs determined by their generalized spectrum

We construct infinite families of graphs that are determined by their generalized spectrum. This construction is based on new formulae for the determinant of the walk matrix of a graph. All graphs constructed here satisfy a certain extremal divisibility condition for the determinant of their walk matrix.     

Constrained Ramsey numbers of graphs

Given two graphs G and H, let f(G,H) denote the minimum integer n such that in every coloring of the edges of K_n, there is either a copy of G with all edges having the same color or a copy of H with all edges having different colors. We show that f(G,H) is finite iff G is a star or H is acyclic. If S and T are trees with s and t edges, respectively, we show that 1+s(t?2)/2≤f(S,T)≤(s?1)(t²+3t). Using constructions from design theory, we establish the exact values, lying near (s?1)(t?1), for f(S,T) when S and T are certain paths or star‐like trees. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 1–16, 2003     

On multicolour noncomplete Ramsey graphs of star graphs

Given graphs , where k≥2, the notation

     

Happy families and completely Ramsey sets

Summary We use games of Kastanas to obtain a new characterization of the classC of all sets that are completely Ramsey with respect to a given happy family . We then combine this with ideas of Plewik to give a uniform proof of various results of Ellentuck, Louveau, Mathias and Milliken concerning the extent ofC . We also study some cardinals that can be associated with the ideal of nowhere -Ramsey sets.Part of this research was done while the author was visiting I.V.I.C. in Caracas in September 1989. The author would like to thank Carlos Di Prisco for his hospitality.     

Strongly regular graphs and finite Ramsey theory

Some connections between strongly regular graphs and finite Ramsey theory are drawn. Let B_n denote the graph K₂+K?_n. If there exists a strongly regular graph with parameters (υ, k, λ, μ), then the Ramsey number r(B_λ+1, B_{υ?2k+μ ?1})?υ+1. We consider the implications of this inequality for both Ramsey theory and the theory of strongly regular graphs.     

Generalised Ramsey numbers for small graphs

The generalised Ramsey number R(G₁, G₂,..., G_k) is defined as the smallest integer n such that, if the edges of K_n, the complete graph on n vertices, are coloured using k colours C₁, C₂,..., C_k, then for some i(1≤i≤k) there is a subgraph G_i of K_n with all of its edges colour C_i. When G₁=G₂=...,G_k=G, we use the more compact notation R_k(G).The generalised Ramsey numbers R_k(G) are investigated for all graphs G having at most four vertices (and no isolates). This extends the work of Chvátal and Harary, who made this investigation in the case k=2.     

Isometric embeddings of subdivided connected graphs into hypercubes

Isometric subgraphs of hypercubes are known as partial cubes. These graphs have first been investigated by Graham and Pollack [R.L. Graham, H. Pollack, On the addressing problem for loop switching, Bell System Technol. J. 50 (1971) 2495-2519; and D. Djokovi?, Distance preserving subgraphs of hypercubes, J. Combin. Theory Ser. B 14 (1973) 263-267]. Several papers followed with various characterizations of partial cubes. In this paper, we determine all subdivisions of a given configuration which can be embedded isometrically in the hypercube. More specifically, we deal with the case where this configuration is a connected graph of order 4, a complete graph of order 5 and the case of a k-fan F_k(k≥3).     

Size Ramsey numbers for small-order graphs

A table of the size Ramsey number or the restricted size Ramsey number for all pairs of graphs with at most four vertices and no isolated vertices is given.     

Some Ramsey numbers for directed graphs

Given k directed graphs G₁,…,G_k the Ramsey number R(G₁,…, G_k) is the smallest integer n such that for any partition (U₁,…,U_k) of the arcs of the complete symmetric directed graph K_n, there exists an integer i such that the partial graph generated by U₁ contains G₁ as a subgraph. In the article we give a necessary and sufficient condition for the existence of Ramsey numbers, and, when they exist an upper bound function. We also give exact values for some classes of graphs. Our main result is: $\text{R(}\text{P}{}_{\mathrm{n}}\text{,….}\text{P}{}_{\mathrm{nk-1}}\text{, G) = n}{}_1\text{…n}{}_{\mathrm{k-1}}\text{(p-1) + 1}$, where G is a hamltonian directed graph with p vertices and $\text{P}{}_{\mathrm{n}}{}_{\mathrm{i}}$ denotes the directed path of length n_t     

Linear Ramsey numbers of sparse graphs

The Ramsey number R(G₁,G₂) of two graphs G₁ and G₂ is the least integer p so that either a graph G of order p contains a copy of G₁ or its complement G^c contains a copy of G₂. In 1973, Burr and Erd?s offered a total of $25 for settling the conjecture that there is a constant c = c(d) so that R(G,G)≤ c|V(G)| for all d‐degenerate graphs G, i.e., the Ramsey numbers grow linearly for d‐degenerate graphs. We show in this paper that the Ramsey numbers grow linearly for degenerate graphs versus some sparser graphs, arrangeable graphs, and crowns for example. This implies that the Ramsey numbers grow linearly for degenerate graphs versus graphs with bounded maximum degree, planar graphs, or graphs without containing any topological minor of a fixed clique, etc. © 2005 Wiley Periodicals, Inc. J Graph Theory