A tripartite-circle drawing of a tripartite graph is in the plane, where each part vertex partition placed on one three disjoint circles, and edges do not cross circles. We present upper lower bounds minimum number crossings drawings $K_{m,n,p}$. In contrast to 1- 2-circle drawings, which may attain Harary-Hill bound, our results imply that balanced restricted 3-circle complete are optimal.

The stability of the equilibria of time-invariant nonlinear dynamical systems with discrete time scale is investigated. We present an elementary proof showing that in the case of a stable equilibrium and continuously di€erentiable state transition function, all eigenvalues of the Jacobian computed at the equilibrium must be inside or on the unit circle. We also demonstrate via numerical example...

Circular-arc graphs are the intersection graphs of open arcs on a circle. Circle graphs are the intersection graphs of chords on a circle. These graph classes have been the subject of much study for many years and numerous interesting results have been reported. Many subclasses of both circular-arc graphs and circle graphs have been defined and different characterizations formulated. In this su...

for any $k in mathbb{n}$, the $k$-subdivision of graph $g$ is a simple graph $g^{frac{1}{k}}$, which is constructed by replacing each edge of $g$ with a path of length $k$. in [moharram n. iradmusa, on colorings of graph fractional powers, discrete math., (310) 2010, no. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $g$ has been introduced as a fractional power of $g$, denoted by ...

The circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant, harmonic, bounded, Dirichlet function. If P is a bounded circle packing in R 2 whose contacts graph is a bounded valence triangulation of a disk, then, with probability 1, the simple random walk on P converges to a limit point. Moreover, in this situation any continuous fun...

An induced matching in a graph G is a set of edges, no two of which meet a common node or are joined by an edge of G; that is, an induced matching is a matching which forms an induced subgraph. Induced matchings in graph G correspond precisely to independent sets of nodes in the square of the line-graph of G, which we denote by [L(G)]. Often, if G has a nice representation as an intersection gr...

In an earlier work, the author extended the AndreevKoebe-Thurston circle packing theorem. Additionally, a polynomial time algorithm for constructing primal-dual circle packings of arbitrary (essentially) 3-connected maps was found. In this note, additional details concerning surfaces of constant curvature 0 (with special emphasis on planar graphs where a slightly different treatment is necessar...

We prove that, for each circle graph H, every with sufficiently large rank-width contains a vertex-minor isomorphic to H.

In their systematic presentations of religious philosophy, the Indian Buddhists consistently defended the position that belief in an eternal creator god who superintends his creation and looks after the concerns of his creatures is a distraction from the central task of the religious life. This was clearly the position taken in the early P~i literature and in the Therav~da philosophy based on t...