619 research outputs found
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Densities of Minor-Closed Graph Families
We define the limiting density of a minor-closed family of simple graphs F to
be the smallest number k such that every n-vertex graph in F has at most
kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting
densities. This set of numbers is countable, well-ordered, and closed; its
order type is at least {\omega}^{\omega}. It is the closure of the set of
densities of density-minimal graphs, graphs for which no minor has a greater
ratio of edges to vertices. By analyzing density-minimal graphs of low
densities, we find all limiting densities up to the first two cluster points of
the set of limiting densities, 1 and 3/2. For multigraphs, the only possible
limiting densities are the integers and the superparticular ratios i/(i+1).Comment: 19 pages, 4 figure
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