Let G be a 2-connected claw-free graph. We show that • if G is N1,1,4-free or N1,2,2-free or Z5-free or P8-free, respectively, then G has a spanning eulerian subgraph (i.e. a spanning connected even subgraph) or its closure is the line graph of a graph in a family of well-defined graphs, • if the minimum degree δ(G) ≥ 3 and G is N2,2,5-free or Z9-free, respectively, then G has a spanning euleri...

We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations. Exact generating functions are obtained for theses graphs with fixed numbers of edges and vertices at given geodesic distances from the origin. Our analysis relies on bijections with labeled trees, in which the labels encode the informat...

Competitive Searching in Polygons Beyond Generalised Streets p. 32 Finding a Shortest Pair of Paths on the Plane with Obstacles and Crossing Areas p. 42 Logspace Verifiers, NC, and NP p. 52 Structure in Average Case Complexity p. 62 Some Geometric Lower Bounds p. 72 The I/O-Complexity of Ordered Binary-Decision Diagram Manipulation p. 82 Two Arc Disjoint Paths in Eulerian Diagraphs p. 92 Findin...

We say that a spanning eulerian subgraph F ⊂ G is a flower in a graph G if there is a vertex u ∈ V (G) (called the center of F ) such that all vertices of G except u are of the degree exactly 2 in F. A graph G has the flower property if every vertex of G is a center of a flower. Kaneko conjectured that G has the flower property if and only if G is hamiltonian. In the present paper we prove this...

Suppose G is a graph embedded in S g with width (also known as edge width) at least 264(2 g ?1). If P V (G) is such that the distance between any two vertices in P is at least 16, then any 5-coloring of P extends to a 5-coloring of all of G. We present similar extension theorems for 6-and 7-chromatic toroidal graphs, for 3-colorable large-width graphs embedded on S g with every face even sided,...

We consider Cheeger Inequalities for general edge-weighted directed graphs. Previously the directed case was considered by Chung for a probability transition matrix corresponding to a strongly connected graph with weights induced by a stationary distribution. An Eulerian property of these special weights reduces these instances to the undirected case, for which recent results on multi-way spect...

An arc colored eulerian multidigraph with l colors is rainbow eulerian if there is an eulerian circuit in which a sequence of l colors repeats. The digraph product that refers the title was introduced by Figueroa-Centeno et al. as follows: let D be a digraph and let Γ be a family of digraphs such that V (F ) = V for every F ∈ Γ. Consider any function h : E(D) −→ Γ. Then the product D⊗hΓ is the ...

In this paper we study a trail routing and a hamiltonian cycle in a class of grid graphs, polycube and polymino. A Spanning closed trail is an eulerian subgraph containing all vertices of a given graph. For general grid graphs we prove that the problem of nding that trail is NP-complete and for a wide subclass of grid graphs, called polymino, we give an optimal algorithm if it exists. For polyc...