1,273 research outputs found
On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings
The problem of establishing the number of perfect matchings necessary to
cover the edge-set of a cubic bridgeless graph is strictly related to a famous
conjecture of Berge and Fulkerson. In this paper we prove that deciding whether
this number is at most 4 for a given cubic bridgeless graph is NP-complete. We
also construct an infinite family of snarks (cyclically
4-edge-connected cubic graphs of girth at least five and chromatic index four)
whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs
were known. It turns out that the family also has interesting
properties with respect to the shortest cycle cover problem. The shortest cycle
cover of any cubic bridgeless graph with edges has length at least
, and we show that this inequality is strict for graphs of .
We also construct the first known snark with no cycle cover of length less than
.Comment: 17 pages, 8 figure
Universal Cycles for Minimum Coverings of Pairs by Triples, with Application to 2-Radius Sequences
A new ordering, extending the notion of universal cycles of Chung {\em et
al.} (1992), is proposed for the blocks of -uniform set systems. Existence
of minimum coverings of pairs by triples that possess such an ordering is
established for all orders. Application to the construction of short 2-radius
sequences is given, with some new 2-radius sequences found through computer
search.Comment: 18 pages, to appear in Mathematics of Computatio
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
- …