For any two graphs F1 and F2, the graph Ramsey number r(F1, F2) is the smallest positive integer N with the property that every graph on at least N vertices contains F1 or its complement contains F2 as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. We determine r(θn,Km) for m = 2, 3, 4 and n > m. More specifically, we establish that r(θn,Km) = (n− 1)(m− 1) ...

A polynomial ideal I ⊆ R[x] is THk-exact if every linear polynomial that is non-negative over VR(I), the real variety of I , is a sum of squares of polynomials of degree at most k modulo I . Lovász recognized that a graph is perfect if and only if the vanishing ideal of the characteristic vectors of its stable sets is TH1-exact, and asked for a characterization of ideals which are TH1-exact. We...

The packing chromatic number χρ(G) of an undirected (respectively, oriented) graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V1, . . . , Vk, in such a way that every two distinct vertices in Vi are at distance (respectively directed distance) greater than i in G for every i, 1 ≤ i ≤ k. The generalized theta graph Θ 1,..., p consis...

Connected sum and trivalent vertex are natural operations on genus 2 spatial graphs and, as with knots, tunnel number behaves in interesting ways under these operations. We prove sharp Scharlemann-Schultens type bounds for the of a composite graph. For Brunnian $\theta$-curve, our result implies that is at least summands, knot case. also version theorem Morimoto knots: m-small graph numbers fac...

Let $\mathcal C_{n}$ be the Catalan monoid on $X_{n}=\{1,\ldots ,n\}$ under its natural order. In this paper, we describe sets of left zero-divisors, right zero-divisors and two sided C_{n}$; their numbers. For $n \geq 4$, define an undirected graph $\Gamma(\mathcal C_{n})$ associated with whose vertices are excluding zero element $\theta$ distinct $\alpha$ $\beta$ joined by edge in case $\alph...

Augmented Lazy Theta*: Applying Graph Abstractions to Trajectory Planning with Speed-Limit Constraints

We continue an investigation initiated by Consani–Marcolli of the relation between the algebraic geometry of p-adic Mumford curves and the noncommutative geometry of graph C-algebras associated to the action of the uniformizing p-adic Schottky group on the Bruhat–Tits tree. We reconstruct invariants of Mumford curves related to valuations of generators of the associated Schottky group, by devel...

Grids with blocked and unblocked cells are often used to represent continuous 2D and 3D environments in robotics and video games. The shortest paths formed by the edges of 8neighbor 2D grids can be up to ≈ 8% longer than the shortest paths in the continuous environment. Theta* typically finds much shorter paths than that by propagating information along graph edges (to achieve short runtimes) w...

In 1979, L. Lovász defined the theta number, a spectral/semidefinite upper bound on the stability number of a graph, which has several remarkable properties (for example, it is exact for perfect graphs). A variant, the inverse theta number, defined recently by the author in a previous work, also constitutes an upper bound on the stability number. In the paper we will describe counterparts of th...

Weak coloring numbers generalize the notion of degeneracy a graph. They were introduced by Kierstead & Yang in context games on graphs. Recently, several connections have been uncovered between weak and various parameters studied graph minor theory its generalizations. In this note, we show that for every fixed $k\geq1$, maximum $r$-th number with simple treewidth $k$ is $\Theta(r^{k-1}\log...