The metamorphosis of lambda-fold 4-wheel systems into lambda-fold bowtie systems

نویسنده

  • Elizabeth J. Billington
چکیده

A 4-wheel is a simple graph on 5 vertices with 8 edges, consisting of a 4-cycle, with a fifth vertex joined to each vertex in the 4-cycle. A A-fold 4-wheel system of order n is an edge-disjoint decomposition of AKn into 4-wheels. If two non-adjacent edges of the 4-cycle are removed, the result is a bowtie (that is, two triangles with a common vertex). In this paper necessary and sufficient conditions are given for the metamorphosis of a A-fold 4-wheel system of order n into a A-fold bowtie system of order n, by retaining the bowtie subgraph from each 4-wheel, and rearranging the disjoint pairs of removed edges from each 4-wheel into further bowties. (There remain three isolated unresolved values of n when A = 2, namely: 24, 72, 88. Currently no 2-fold 4-wheel systems of these orders are known.) 1 Introduction and necessary conditions Let G and H be simple graphs, and let AH denote the graph H with each of its edges replicated A times. A A-fold G-system of AH is a pair (X, K) where X is the vertex set of Hand K is a collection of isomorphic copies of the graph G whose edges partition the edges of AH. If H is a complete graph K n , we refer to such a A-fold G-system as one of order n. Also if A = 1, we drop the term "I-fold". A 4-wheel G is a simple graph with 5 vertices {c, aI, a2, a3, a4} and 8 edges i 1,2,3 or 4 (subscript addition modulo 4). A bowtie G' is a simple graph with 5 vertices and 6 edges, consisting of two triangles sharing one common vertex. If the two triangles have vertices {a, b, c} and

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عنوان ژورنال:
  • Australasian J. Combinatorics

دوره 22  شماره 

صفحات  -

تاریخ انتشار 2000