322 research outputs found
On star forest ascending subgraph decomposition
The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with (n+12) edges admits an edge decomposition G=H1¿¿¿Hn such that Hi has i edges and it is isomorphic to a subgraph of Hi+1, i=1,…,n-1. We show that every bipartite graph G with (n+12) edges such that the degree sequence d1,…,dk of one of the stable sets satisfies dk-i=n-ifor each0=i=k-1, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.Peer ReviewedPostprint (author's final draft
On star forest ascending subgraph decomposition
The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with (n+12) edges admits an edge decomposition G=H1¿¿¿Hn such that Hi has i edges and it is isomorphic to a subgraph of Hi+1, i=1,…,n-1. We show that every bipartite graph G with (n+12) edges such that the degree sequence d1,…,dk of one of the stable sets satisfies dk-i=n-ifor each0=i=k-1, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.Peer ReviewedPostprint (author's final draft
On the bordification of outer space
We give a simple construction of an equivariant deformation retract of Outer
space which is homeomorphic to the Bestvina-Feighn bordification. This results
in a much easier proof that the bordification is (2n-5)-connected at infinity,
and hence that is a virtual duality group.Comment: Accepted version, to appear in the Journal of the London MS. Section
7, giving the homeomorphism to the Bestvina-Feighn bordification, has been
substantially revise
Ascending Subgraph Decomposition
A typical theme for many well-known decomposition problems is to show that
some obvious necessary conditions for decomposing a graph into copies are also sufficient. One such problem was posed in 1987, by Alavi,
Boals, Chartrand, Erd\H{o}s, and Oellerman. They conjectured that the edges of
every graph with edges can be decomposed into subgraphs such that each has edges and is isomorphic to a subgraph
of . In this paper we prove this conjecture for sufficiently large
Rectangle Visibility Numbers of Graphs
Very-Large Scale Integration (VLSI) is the problem of arranging components on the surface of a circuit board and developing the wired network between components. One methodology in VLSI is to treat the entire network as a graph, where the components correspond to vertices and the wired connections correspond to edges. We say that a graph G has a rectangle visibility representation if we can assign each vertex of G to a unique axis-aligned rectangle in the plane such that two vertices u and v are adjacent if and only if there exists an unobstructed horizontal or vertical channel of finite width between the two rectangles that correspond to u and v. If G has such a representation, then we say that G is a rectangle visibility graph.
Since it is likely that multiple components on a circuit board may represent the same electrical node, we may consider implementing this idea with rectangle visibility graphs. The rectangle visibility number of a graph G, denoted r(G), is the minimum k such that G has a rectangle visibility representation in which each vertex of G corresponds to at most k rectangles. In this thesis, we prove results on rectangle visibility numbers of trees, complete graphs, complete bipartite graphs, and (1,n)-hilly graphs, which are graphs where there is no path of length 1 between vertices of degree n or more
Actions of Maximal Growth
We study acts and modules of maximal growth over finitely generated free
monoids and free associative algebras as well as free groups and free group
algebras. The maximality of the growth implies some other specific properties
of these acts and modules that makes them close to the free ones; at the same
time, we show that being a strong "infiniteness" condition, the maximality of
the growth can still be combined with various finiteness conditions, which
would normally make finitely generated acts finite and finitely generated
modules finite-dimensional
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