172. Three conics connected with a triangle

Ramsey Number of a Connected Triangle Matching

We determine the 2-color Ramsey number of a connected triangle matching c(nK3) which is a graph with n vertex disjoint triangles plus (at least n − 1) further edges that keep these triangles connected. We obtain that R(c(nK3), c(nK3)) = 7n − 2, somewhat larger than in the classical result of Burr, Erdős and Spencer for a triangle matching, R(nK3, nK3) = 5n. The motivation is to determine the Ra...

Deficiencies of connected regular triangle free graphs

Let G be a finite simple graph having a maximum matching M. The deficiency def(G) of G is the number of M-unsaturated vertices in G. In an earlier paper we determined an upper bound for def(G) when G is regular and connected. This upper bound is in general not sharp when G is triangle free. In this paper we study the case when G is triangle free and r-regular. We present an upper bound for defC...

Connected, Bounded Degree, Triangle Avoidance Games

We consider variants of the triangle-avoidance game first defined by Harary and rediscovered by Hajnal a few years later. A graph game begins with two players and an empty graph on n vertices. The two players take turns choosing edges within Kn, building up a simple graph. The edges must be chosen according to a set of restrictions R. The winner is the last player to choose an edge that does no...

Liquids with conics

A Family of Conics and Three Special Ruled Surfaces

In [5] the authors presented a family F = {ck | k ∈ R} of conics. The conics ck are gained by offsetting from a given conic c0 with proportional distance functions kδ(t). We investigate certain properties of F and give the correct version of a result claimed in [5]: The distance function is unique (up to a constant factor) only if c0 is not a parabola. Furthermore we deal with the surfaces that...