The number of triangles in a K4-free graph

Making a K4-free graph bipartite

We show that every K4-free graph G with n vertices can be made bipartite by deleting at most n/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős.

List-colouring the square of a K4-minor-free graph

Let G be a K4-minor-free graph with maximum degree . It is known that if ∈ {2, 3} then G2 is ( + 2)-degenerate, so that (G2) ch(G2) + 3. It is also known that if 4 then G2 is ( 3 2 + 1)-degenerate and (G2) 3 2 + 1. It is proved here that if 4 then G2 is 3 2 -degenerate and ch(G2) 3 2 + 1. These results are sharp. © 2007 Elsevier B.V. All rights reserved.

The Maximum Number of Edges in a Strongly Multiplicative Graph

Uniform Number of a Graph

We introduce the notion of uniform number of a graph. The  uniform number of a connected graph $G$ is the least cardinality of a nonempty subset $M$ of the vertex set of $G$ for which the function $f_M: M^crightarrow mathcal{P}(X) - {emptyset}$ defined as $f_M(x) = {D(x, y): y in M}$ is a constant function, where $D(x, y)$ is the detour distance between $x$ and $y$ in $G$ and $mathcal{P}(X)$ ...

determination of maximal singularity free zones in the workspace of parallel manipulator

due to the limiting workspace of parallel manipulator and regarding to finding the trajectory planning of singularity free at workspace is difficult, so finding a best solution that can develop a technique to determine the singularity-free zones in the workspace of parallel manipulators is highly important. in this thesis a simple and new technique are presented to determine the maximal singula...