András Gyárfás

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

AbstractWe present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least

Vertex covers by monochromatic pieces - A survey of results and problems

tfrac{{(1 + varepsilon )3|V(G)|}}n{4}

Coverings by Few Monochromatic Pieces: A Transition Between Two Ramsey Problems

can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that

Chromatic Ramsey number of acyclic hypergraphs

bar G

Covering complete partite hypergraphs by monochromatic components

does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| − c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C4)=1, which is best possible.

Ramsey number of paths and connected matchings in Ore-type host graphs

The chromatic gap is the difference between the chromatic number and the clique number of a graph. Here we investigate gap(n), the maximum chromatic gap over graphs on n vertices. Can the extremal graphs be explored? While computational problems related to the chromatic gap are hopeless, an interplay between Ramsey-theory and matching theory leads to a simple and (almost) exact formula for gap(n) in terms of Ramsey-numbers. For our purposes it is more convenient to work with the covering gap, the difference between the clique cover number and stability number of a graph and this is what we call the gap of a graph. Notice that the well-studied family of perfect graphs are the graphs whose induced subgraphs have gap zero. The maximum of the (covering) gap and the chromatic gap running on all induced subgraphs will be called perfectness gap. Using @a(G) for the cardinality of a largest stable (independent) set of a graph G, we define @a(n)=min@a(G) where the minimum is taken over triangle-free graphs on n vertices. It is easy to observe that @a(n) is essentially an inverse Ramsey function, defined by the relation R(3,@a(n))==0, we denote by s(t) the smallest order of a graph with gap t and we call a graph is t-extremal if it has gap t and order s(t). Equivalently, s(t) is the smallest order of a graph with perfectness gap equal to t. It is tempting to conjecture that s(t)=5t with equality for the graph tC5. However, for t>=3 the graph tC5 has gap t but it is not gap-extremal (although gap-critical). We shall prove that s(3)=13, s(4)=17 and s(5)@?{20,21}. Somewhat surprisingly, after the uncertain values s(6)@?{23,24,25}, s(7)@?{26,27,28}, s(8)@?{29,30,31}, s(9)@?{32,33} we can show that s(10)=35. On the other hand we can easily show that s(t) is asymptotically equal to 2t, that is, gap(n) is asymptotic to n/2. According to our main result the gap is actually equal to @?n/2@?-@a(n), unless n is in an interval [R,R+14] where R is a Ramsey-number, and if this exception occurs the gap may be larger than this value by only a small constant (at most 3). Our study provides some new properties of Ramsey graphs them selves: it shows that triangle-free Ramsey graphs have high matchability and connectivity properties, leading possibly to new bounds on Ramsey-numbers.