Related papers: Region crossing change on planar trivalent graphs
Region crossing change is a local operation on link diagrams. The behavior of region crossing change on $S^2$ is well understood. In this paper, we study the behavior of (modified) region crossing change on higher genus surfaces.
A region crossing change at a region of a spatial-graph diagram is a transformation changing every crossing on the boundary of the region. In this paper, it is shown that every spatial graph consisting of theta-curves can be unknotted by…
We introduce a local move on a link diagram named a region freeze crossing change which is close to a region crossing change, but not the same. We study similarity and difference between region crossing change and region freeze crossing…
A region crossing change is a local transformation on spatial graph diagrams switching the over/under relations at all the crossings on the boundary of a region. In this paper, we show that a spatial graph of a planar graph is unknottable…
Region crossing change is a local transformation on a knot or link diagram. We show that a region crossing change on a knot diagram is an unknotting operation, and we define the region unknotting numbers for a knot diagram and a knot.
We introduce the triple crossing number, a variation of crossing number, of a graph, which is the minimal number of crossing points in all drawings with only triple crossings of the graph. It is defined to be zero for a planar graph, and to…
In this paper, we prove that region crossing change on a link diagram is an unknotting operation if and only if the link is proper. A description of the behavior of region crossing change on link diagrams is given. Furthermore we also…
We give a description of local and global moves on a class of locally planar trivalent graphs and we show that it contains $\lambda$-Scale calculus, therefore in particular untyped lambda calculus. Surprisingly, the beta reduction rule…
The purpose of this article is to give a preliminary clarification on the relation between crossing number and crossing change. With a main focus on the span of X polynomial, we prove that, as our theorem claims, the crossing number of the…
We construct a state model for the two-variable Kauffman polynomial using planar trivalent graphs. We also use this model to obtain a polynomial invariant for a certain type of trivalent graphs embedded in three-dimensional space.
We survey algorithms and bounds for constructing planar drawings of graphs in small area.
We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of…
In this paper, we study some properties of multivariate gamma function and zonal polynomials.
Human perception of graph drawings is influenced by a variety of impact factors for which quality measures are used as a proxy indicator. The investigation of those impact factors and their effects is important to evaluate and improve…
We discuss the cross-ratio map of planar quadrilateral linkages, also in the case when one of the links is telescopic. Most of our results are valid for a planar quadrilateral linkage with generic lengths of the sides. In particular, we…
We improve the estimates of the subgraph probabilities in a random regular graph. Using the improved results, we further improve the limiting distribution of the number of triangles in random regular graphs.
In this paper, we introduce a generalized concept of vertex transitivity in graphs called generalized vertex transitivity. We put forward a new invariant called transitivity number of a graph. The value of this invariant in different…
We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees.
We prove that sparse string graphs in a fixed surface have linear expansion. We extend this result to the more general setting of sparse region intersection graphs over any proper minor-closed class. The proofs are combinatorial and…
We present, to the best of the authors' knowledge, all known results for the (planar) crossing numbers of specific graphs and graph families. The results are separated into various categories; specifically, results for general graph…