The Size Ramsey Number of Graphs with Bounded Treewidth

On the Number of Labeled Graphs of Bounded Treewidth

We focus on counting the number of labeled graphs on n vertices and treewidth at most k (or equivalently, the number of labeled partial k-trees), which we denote by Tn,k. So far, only the particular cases Tn,1 and Tn,2 had been studied. We show that

On Size Ramsey Numbers of Graphs with Bounded Degree

Answering a question of J. Beck B2], we prove that there exists a graph G on n vertices with maximum degree three and the size Ramsey number ^ r(G) cn(log n) where and c are positive constants. For graphs G and F, write F ! G to mean that if the edges of F are colored by red and blue, then F contains a monochromatic copy of G. Erd} os, Faudree, Rousseau and Schelp EFRS] were the rst to consider...

Recoloring bounded treewidth graphs

Let k be an integer. Two vertex k-colorings of a graph are adjacent if they differ on exactly one vertex. A graph is k-mixing if any proper k-coloring can be transformed into any other through a sequence of adjacent proper k-colorings. Any graph is (tw + 2)-mixing, where tw is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two (tw + 2)-colorings is a...

The size Ramsey number of short subdivisions of bounded degree 1 graphs ∗

5 For graphs G and F , write G → (F )` if any coloring of the edges of G with ` colors yields 6 a monochromatic copy of the graph F . Let positive integers h and d be given. Suppose S 7 is obtained from a graph S with s vertices and maximum degree d by subdividing its edges h 8 times (that is, by replacing the edges of S by paths of length h+ 1). We prove that there exits 9 a graph G with no mo...

Dynamic Programming on Graphs with Bounded Treewidth