Consider an embedding of a graph G in a surface S (map). Assume that the diierence splits into connected components (countries), each one homeomorphic to an open disk. (It follows from this assumption that graph G must be connected). Introduce a graph G dual to G realizing the neighbor relations among countries. The graphs G and G have the same set of edges. More precisely, there is a natural o...

We present a simple depth-rst search strategy for exploring (constructing) an unknown strongly connected graph G with m edges and n vertices by traversing at most min(mn; dn 2 + m) edges. Here, d is the minimum number of edges needed to add to G to make it Eulerian. This parameter d is known as the deeciency of a graph and was introduced by Kutten Kut88]. It was conjectured that graphs with low...

We consider the travelling salesman problem on metrics which can be viewed as the shortest path metric of an undirected graph with unit edge-lengths. Finding a TSP tour in such a metric is then equivalent to finding a connected Eulerian subgraph in the underlying graph. Since the length of the tour is the number of edges in this Eulerian subgraph our problem can equivalently be stated as follow...

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph n vertices can be decomposed into at most ? 2 ? paths, while a conjecture Hajós states Eulerian ? ? 1 ? cycles. The Erd?s–Gallai O ( ) cycles edges. show if G is sufficiently large with linear minimum degree, then following hold. (i) + o p...

The evolutionary theory, " evolution by duplication " , originally proposed by Susumu Ohno in 1970, can now be verified with the available genome sequences. Recently, several mathematical models have been proposed to explain the topol-ogy of protein interaction networks that have also implemented the idea of " evolution by duplication ". The power law distribution with its " hubby " topology (e...

The set of integral ows on a nite graph ? is naturally an integral lattice 1 (?) in the Euclidean space Ker((1) of harmonic real-valued functions on the edge set of ?. Various properties of ? (bipartite character, girth, complexity, separability) are shown to correspond to properties of 1 (?) (parity, minimal norm, determinant, decomposability). The dual lattice of 1 (?) is identiied to the int...

9:30 10:30 Bruce Reed How many edges force an H-minor? 10:30 11:00 Coffee Break 11:00 11:20 Chun-Hung Liu Well-quasi-ordering graphs by the topological minor relation 11:25 11:45 Jean-Florent Raymond An edge variant of Erdős-Pósa property 11:50 12:10 Irene Muzi Subdivisions in 4-connected graphs of large tree-width 12:10 3:30 Lunch and discussion 3:30 4:30 Dan Kral FO limits of trees 4:30 5:00 ...

1 1 General introduction 3 1.1 Combinatorial property testing . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Property testing in the dense graph (and hypergraph) model . . . . 4 1.1.2 Testing graph properties in other models . . . . . . . . . . . . . . . 5 1.1.3 Testing massively parameterized properties . . . . . . . . . . . . . . 5 1.2 Probabilistically checkable proofs of proximit...

The set of integral ows on a nite graph ? is naturally an integral lattice 1 (?) in the Euclidean space Ker((1) of harmonic real-valued functions on the edge set of ?. Various properties of ? (bipartite character, girth, complexity, separability) are shown to correspond to properties of 1 (?) (parity, minimal norm, determinant, decomposability). The dual lattice of 1 (?) is identiied to the int...

The goal of genome assembly is to reconstruct contiguous portions of a genome (known as contigs) given short reads of DNA sequence obtained in a sequencing experiment. De Bruijn graphs are constructed by finding overlaps of length k − 1 between all substrings of length k from the reads, resulting in a graph where the correct reconstruction of the genome is given by one of the many possible Eule...