The vertex arboricity $rho(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ can be partitioned so that each subset induces an acyclic graph‎. ‎A graph $G$ is called list vertex $k$-arborable if for any set $L(v)$ of cardinality at least $k$ at each vertex $v$ of $G$‎, ‎one can choose a color for each $v$ from its list $L(v)$ so that the subgraph induced by ev...

We present an algorithm for finding shortest surface non-separating cycles in graphs embedded on surfaces in O(gV 3/2 log V + gV ) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V ), this represents a considerable improvement over previous results by Thomassen, and Erickson and HarPeled. We also give algorithms to find a shortest non-contractibl...

A toroidalization of a dominant morphism $varphi: Xto Y$ of algebraic varieties over a field of characteristic zero is a toroidal lifting of $varphi$ obtained by performing sequences of blow ups of nonsingular subvarieties above $X$ and $Y$. We give a proof of toroidalization of locally toroidal morphisms of 3-folds.

Let Φ be an embedding of graph G in a surface S. If there exists a subset K of S bounded by a subgraph of G which contains all the vertices of G, then K is called a spanning subset of Φ. Examples of spanning subsets include spanning discs, spanning annuli with some number of holes (called planarizing sets in some papers). A spanning subset may provide a simpler structure but still contain enoug...

Among different goals defined in vehicle design process, fuel consumption (FC) is one of the most important objectives, which significantly has taken into account lately, both by the customers and vehicle manufacturers. One of the significant parameters which impacts the vehicle FC is the efficiency of vehicle's power train. In this paper, a half-toroidal continuously variable transmission (CVT...

We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree tc(g) associated with any 2-connected graph g, whose white vertices are the 3-components of g (3-connected components or polygons) and black vertices are bonds a...

A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two k-colorings are k-equivalent if we can transform one into other using swaps. The triangulated toroidal grid, T[m×n], is formed from (a embedding of) Cartesian product of Cm and Cn by adding parallel diagonals inside all 4-faces. Mohar Salas showed that not 4-colorings T[m×n] 4-equivalent....

In this paper, we show that seven pages are sufficient for a book embedding of any toroidal graph.

In [9] Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in Rd. Jackson Jordán [10] confirmed these also sufficient R2, giving combinatorial characterization of graphs whose realizations R2 rigid. this paper, we establish analogues results infinite periodic frameworks under fixed latti...