Edge-disjoint Triangles Research Articles

The spectrum for 2-perfect bowtie systems

A bowtie is a pair of edge disjoint triangles of K v with a common vertex. A bowtie system is an edge disjoint decomposition of K v into bowties. A bowtie system is 2-perfect if it has the additional property that each bowtie can be replaced by exactly one of its distance 2 graphs so that the resulting collection of bowties is also a bowtie system. We show that the spectrum of 2-perfect bowtie systems is precisely the set of all n1 or 9 (mod 12), with the possible exceptions of n=69 and 81. We also solve the same problem for K v ⧹ K 3 . That is, we show that a 2-perfect decomposition of K v ⧹ K 3 into bowties exists if and only if v3 or 7 (mod 12).

On reliable graphs with static routing plans

We consider routing plans on graphs with two entries for each origination-destination pair. It is shown that graphs whose edge set can be covered by edge-disjoint triangles allow reliable networks. We conclude with some remarks on optimally reliable graphs.

A conjecture on triangles of graphs

We give various results related to the following conjecture: If a graphG does not contain more thank pairwise edge-disjoint triangles, then there exists a set of at most 2k edges that meets all triangles ofG.

On complete subgraphs of different orders

LetSbe a set and let {X1, …,Xn} =be a family of distinct subsets ofS. Theintersection graphΩ() ofhas vertex set {X1, …,Xn} andXiXj(i≠j) is an edge of Ω() if and only ifXi∩Xi≠ Ø (c.f. (6)). It is easily seen, (7), that every graph is an intersection graph, in other words every graph can be represented by subsets ofa set. Moreover, it was proved by Erdös, Goodman and Pósa (5) that if a graph hasn≥ 4 vertices then one can find a representing set with at most [n2/4] elements. This assertion is an immediate consequence of the result, (5), that the edges of a graph withn≥ 1 vertices can be covered with at most [n2/4] edge disjoint triangles and edges. We say that a graphGiscoveredwith the subgraphsG1, …,Gk, if everyedgeofGis in at least oneGi. One of the aims of this note is to prove an extension of this result, pro-posed by Erdös (4).

Degrees of Vertices in a Friendship Graph

AbstractA friendship graph is a graph in which every two distinct vertices have exactly one common adjacent vertex (called a neighbour). Finite friendship graphs have been characterized by Erdós, Rényi and Sós [2]: Each finite friendship graph Fn which consists of n edge disjoint triangles such that all n>1 triangles have one vertex in common (F1 is a triangle i.e. the complete graph with three vertices). Thus Fn has 2n+1 vertices, 2n of them being of degree two and the remaining one (the common vertex of n triangles if n>1) being of degree 2n.

On the Maximal Number of Edges in Graphs with a Given Number of Edge-Disjoint Triangles

On the Maximal Number of Edges in Graphs with a Given Number of Edge-Disjoint Triangles