The planar Ramsey number PR (H1, H2) is the smallest integer n such that any planar graph on n vertices contains a copy of H1 or its complement contains a copy of H2. It is known that the Ramsey number R(K4 − e, K6) = 21, and the planar Ramsey numbers PR(K4 − e, Kl) for l ≤ 5 are known. In this paper, we give the lower bounds on PR (K4 − e, Kl) and determine the exact value of PR (K4 − e, K6).

‎An  oriented perfect path double cover (OPPDC) of a‎ ‎graph $G$ is a collection of directed paths in the symmetric‎ ‎orientation $G_s$ of‎ ‎$G$ such that‎ ‎each arc‎ ‎of $G_s$ lies in exactly one of the paths and each‎ ‎vertex of $G$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete‎ ‎Math‎. ‎276 (2004) 287-294) conjectured that ...

Classical Multi-color Ramsey Theory pertains to the existence of monochromatic subsets of structured multicolored sets such that the subsets have a given property or structure. This paper examines a generalization of Ramsey theory that allows the subsets to have specified groupings of colors. By allowing more than one color in subsets, the corresponding minimal sets for finite cases tend to be ...

1 The Ramsey number R(G1,G2) is the smallest integer p such that for any graph G on p vertices 2 either G contains G1 or G contains G2, where G denotes the complement of G. In this paper, some 3 new bounds with two parameters for the Ramsey number R(G1,G2), under some assumptions, are 4 obtained. Especially, we prove that R(K6 − e, K6) ≤ 116 and R(K6 − e, K7) ≤ 202, these improve 5 the two uppe...

In this paper we present three Ramsey-type results, which we derive from a simple and yet powerful lemma, proved using probabilistic arguments. Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph Ks on s vertices. More than 40 years ago Erdős and Rogers posed the problem of estimating the maximum size of a subset of G without a copy of the complete...

For two given graphs G and H the planar Ramsey number PR(G,H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G or its complement contains a copy H . By studying the existence of subhamiltonian cycles in complements of sparse graphs, we determine all planar Ramsey numbers for pairs of cycles.

A bowtie is a graph consisting of two triangles with one vertex identified. We show that the class of all (countable) graphs not containing a bowtie as a subgraph have a Ramsey lift (expansion). This is the first non-trivial Ramsey class with a non-trivial algebraic closure.

s Elgersburg 2011 Rainbow Cycles in Cube Graphs Jens-P. Bode (Technische Universität Braunschweig) Joint work with A. Kemnitz and S. Struckmann A graph G is called rainbow with respect to an edge coloring if no two edges of G have the same color. Given a host graph H and a guest graph G ⊆ H, an edge coloring of H is called G-anti-Ramsey if no subgraph of H isomorphic to G is rainbow. The anti-R...

In this paper we translate Ramsey-type problems into the language of decomposable hereditary properties of graphs. We prove a distribu-tive law for reducible and decomposable properties of graphs. Using it we establish some values of graph theoretical invariants of decompos-able properties and show their correspondence to generalized Ramsey numbers.