8,274 research outputs found
The Erdős-Ko-Rado properties of various graphs containing singletons
Let G=(V,E) be a graph. For r≥1, let be the family of independent vertex r-sets of G. For vV(G), let denote the star . G is said to be r-EKR if there exists vV(G) such that for any non-star family of pair-wise intersecting sets in . If the inequality is strict, then G is strictly r-EKR.
Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if GΓ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2r is no larger than μ(G), the size of a smallest maximal independent vertex set of G, then G is r-EKR, and strictly so if 2r<μ(G). We show that in fact, if GΓ and 2r is no larger than the independence number of G, then G is r-EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ′Γ. We also confirm the conjecture for graphs in an even larger set Γ″Γ′
A Coupled AKNS-Kaup-Newell Soliton Hierarchy
A coupled AKNS-Kaup-Newell hierarchy of systems of soliton equations is
proposed in terms of hereditary symmetry operators resulted from Hamiltonian
pairs. Zero curvature representations and tri-Hamiltonian structures are
established for all coupled AKNS-Kaup-Newell systems in the hierarchy.
Therefore all systems have infinitely many commuting symmetries and
conservation laws. Two reductions of the systems lead to the AKNS hierarchy and
the Kaup-Newell hierarchy, and thus those two soliton hierarchies also possess
tri-Hamiltonian structures.Comment: 15 pages, late
On dynamic monopolies of graphs with general thresholds
Let be a graph and be an
assignment of thresholds to the vertices of . A subset of vertices is
said to be dynamic monopoly (or simply dynamo) if the vertices of can be
partitioned into subsets such that and for any
each vertex in has at least neighbors in
. Dynamic monopolies are in fact modeling the irreversible
spread of influence such as disease or belief in social networks. We denote the
smallest size of any dynamic monopoly of , with a given threshold
assignment, by . In this paper we first define the concept of a
resistant subgraph and show its relationship with dynamic monopolies. Then we
obtain some lower and upper bounds for the smallest size of dynamic monopolies
in graphs with different types of thresholds. Next we introduce
dynamo-unbounded families of graphs and prove some related results. We also
define the concept of a homogenious society that is a graph with probabilistic
thresholds satisfying some conditions and obtain a bound for the smallest size
of its dynamos. Finally we consider dynamic monopoly of line graphs and obtain
some bounds for their sizes and determine the exact values in some special
cases
Matchings and Hamilton Cycles with Constraints on Sets of Edges
The aim of this paper is to extend and generalise some work of Katona on the
existence of perfect matchings or Hamilton cycles in graphs subject to certain
constraints. The most general form of these constraints is that we are given a
family of sets of edges of our graph and are not allowed to use all the edges
of any member of this family. We consider two natural ways of expressing
constraints of this kind using graphs and using set systems.
For the first version we ask for conditions on regular bipartite graphs
and for there to exist a perfect matching in , no two edges of which
form a -cycle with two edges of .
In the second, we ask for conditions under which a Hamilton cycle in the
complete graph (or equivalently a cyclic permutation) exists, with the property
that it has no collection of intervals of prescribed lengths whose union is an
element of a given family of sets. For instance we prove that the smallest
family of -sets with the property that every cyclic permutation of an
-set contains two adjacent pairs of points has size between
and . We also give bounds on the general version of this problem
and on other natural special cases.
We finish by raising numerous open problems and directions for further study.Comment: 21 page
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