We consider testing directed graphs for being Eulerian in the orientation model introduced in [15]. Despite the local nature of the property of being Eulerian, it turns out to be significantly harder for testing than other properties studied in the orientation model. We show a non-constant lower bound on the query complexity of 2-sided tests and a linear lower bound on the query complexity of 1...

The combinatorial Hopf algebra on building sets BSet extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the corresponding hypergraph. By passing from graphs to building sets, we construct a sequence of symmetric functions associated to a graph. From the generalized DehnSomm...

We consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although it is known that this can be achieved in polynomial time for any graph, the algorithm studied here is more natural in the context of planar Eulerian graphs. We analyse the mixing time of a Markov chain on the Eulerian orientations of a planar gra...

Algebraic techniques are used to find several new combinatorial interpretations for valuations of the Martin polynomial, M(G; s), for unoriented graphs. The Martin polynomial of a graph, introduced by Martin in his 1977 thesis, encodes information about the families of closed paths in Eulerian graphs. The new results here are found by showing that the Martin polynomial is a translation of a uni...

The opinion forming process in a social network could be naturally modeled as an opinion influencing and updating dynamics. This already attracted researchers interest a while ago in mathematical sociology, and recently in theoretical computer science. In so-called opinion formation games, when underlying networks are directed, a bounded price of anarchy is only known for weighted Eulerian grap...

For an integer s > 0 and for u, v ∈ V (G) with u ≠ v, an (s; u, v)-trail-system of G is a subgraphH consisting of s edge-disjoint (u, v)-trails. A graph is supereulerianwithwidth s if for any u, v ∈ V (G) with u ≠ v, G has a spanning (s; u, v)-trail-system. The supereulerian width μ(G) of a graph G is the largest integer s such that G is supereulerian with width k for every integer kwith 0 ≤ k ...

We consider testing directed graphs for being Eulerian in the orientation model introduced in [15]. Despite the local nature of the property of being Eulerian, it turns out to be significantly harder for testing than other properties studied in the orientation model. We show a superconstant lower bound on the query complexity of 2-sided tests and a linear lower bound on the query complexity of ...

We introduce the notion of an Eulerian graph being strongly evencycle decomposable, and note that a few basic classes have this property. Our main results are that several fundamental composition operations that preserve the property of being Eulerian also preserve the property of being strongly even-cycle decomposable. As an application, our composition operations easily imply that all Euleria...