6,140 research outputs found
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
Problems in extremal graph theory
We consider a variety of problems in extremal graph and set theory.
The {\em chromatic number} of , , is the smallest integer
such that is -colorable.
The {\it square} of , written , is the supergraph of in which also
vertices within distance 2 of each other in are adjacent.
A graph is a {\it minor} of if
can be obtained from a subgraph of by contracting edges.
We show that the upper bound for
conjectured by Wegner (1977) for planar graphs
holds when is a -minor-free graph.
We also show that is equal to the bound
only when contains a complete graph of that order.
One of the central problems of extremal hypergraph theory is
finding the maximum number of edges in a hypergraph
that does not contain a specific forbidden structure.
We consider as a forbidden structure a fixed number of members
that have empty common intersection
as well as small union.
We obtain a sharp upper bound on the size of uniform hypergraphs
that do not contain this structure,
when the number of vertices is sufficiently large.
Our result is strong enough to imply the same sharp upper bound
for several other interesting forbidden structures
such as the so-called strong simplices and clusters.
The {\em -dimensional hypercube}, ,
is the graph whose vertex set is and
whose edge set consists of the vertex pairs
differing in exactly one coordinate.
The generalized Tur\'an problem asks for the maximum number
of edges in a subgraph of a graph that does not contain
a forbidden subgraph .
We consider the Tur\'an problem where is and
is a cycle of length with .
Confirming a conjecture of Erd{\H o}s (1984),
we show that the ratio of the size of such a subgraph of
over the number of edges of is ,
i.e. in the limit this ratio approaches 0
as approaches infinity
Disjoint Cycles in Eulerian Digraphs and the Diameter of Interchange Graphs
AbstractLet R=(r1, …, rm) and S=(s1, …, sn) be nonnegative integral vectors with ∑ri=∑sj. Let A(R, S) denote the set of all m×n {0, 1}-matrices with row sum vector R and column sum vector S. Suppose A(R, S)≠∅. The interchange graphG(R, S) of A(R, S) was defined by Brualdi in 1980. It is the graph with all matrices in A(R, S) as its vertices and two matrices are adjacent provided they differ by an interchange matrix. Brualdi conjectured that the diameter of G(R, S) cannot exceed mn/4. A digraph G=(V, E) is called Eulerian if, for each vertex u∈V, the outdegree and indegree of u are equal. We first prove that any bipartite Eulerian digraph with vertex partition sizes m, n, and with more than (17−1)mn/4 (≈0.78mn) arcs contains a cycle of length at most 4. As an application of this, we show that the diameter of G(R, S) cannot exceed (3+17)mn/16 (≈0.445mn). The latter result improves a recent upper bound on the diameter of G(R, S) by Qian. Finally, we present some open problems concerning the girth and the maximum number of arc-disjoint cycles in an Eulerian digraph
Bounds for phylogenetic network space metrics
Phylogenetic networks are a generalization of phylogenetic trees that allow for representation of reticulate evolution. Recently, a space of unrooted phylogenetic networks was introduced, where such a network is a connected graph in which every vertex has degree 1 or 3 and whose leaf-set is a fixed set X of taxa. This space, denoted N(X)N(X) , is defined in terms of two operations on networks—the nearest neighbor interchange and triangle operations—which can be used to transform any network with leaf set X into any other network with that leaf set. In particular, it gives rise to a metric d on N(X)N(X) which is given by the smallest number of operations required to transform one network in N(X)N(X) into another in N(X)N(X) . The metric generalizes the well-known NNI-metric on phylogenetic trees which has been intensively studied in the literature. In this paper, we derive a bound for the metric d as well as a related metric dNNIdNNI which arises when restricting d to the subset of N(X)N(X) consisting of all networks with 2(|X|−1+i)2(|X|−1+i) vertices, i≥1i≥1 . We also introduce two new metrics on networks—the SPR and TBR metrics—which generalize the metrics on phylogenetic trees with the same name and give bounds for these new metrics. We expect our results to eventually have applications to the development and understanding of network search algorithms
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