7,224 research outputs found
Edge-disjoint spanning trees and eigenvalues of regular graphs
Partially answering a question of Paul Seymour, we obtain a sufficient
eigenvalue condition for the existence of edge-disjoint spanning trees in a
regular graph, when . More precisely, we show that if the second
largest eigenvalue of a -regular graph is less than
, then contains at least edge-disjoint spanning
trees, when . We construct examples of graphs that show our
bounds are essentially best possible. We conjecture that the above statement is
true for any .Comment: 4 figure
-covering red and blue points in the plane
We say that a finite set of red and blue points in the plane in general
position can be -covered if the set can be partitioned into subsets of
size , with points of one color and point of the other color, in
such a way that, if at each subset the fourth point is connected by
straight-line segments to the same-colored points, then the resulting set of
all segments has no crossings. We consider the following problem: Given a set
of red points and a set of blue points in the plane in general
position, how many points of can be -covered? and we prove
the following results:
(1) If and , for some non-negative integers and ,
then there are point sets , like -equitable sets (i.e.,
or ) and linearly separable sets, that can be -covered.
(2) If , and the points in are in convex position,
then at least points can be -covered, and this bound is tight.
(3) There are arbitrarily large point sets in general position,
with , such that at most points can be -covered.
(4) If , then at least points of
can be -covered. For , there are too many red points and at
least of them will remain uncovered in any -covering.
Furthermore, in all the cases we provide efficient algorithms to compute the
corresponding coverings.Comment: 29 pages, 10 figures, 1 tabl
K1,3-covering red and blue points in the plane
We say that a finite set of red and blue points in the plane in general position can be K1, 3-covered if the set can be partitioned into subsets of size 4, with 3 points of one color and 1 point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set R of r red points and a set B of b blue points in the plane in general position, how many points of R ¿ B can be K1, 3-covered? and we prove the following results: (1) If r = 3g + h and b = 3h + g, for some non-negative integers g and h, then there are point sets R ¿ B, like {1, 3}-equitable sets (i.e., r = 3b or b = 3r) and linearly separable sets, that can be K1, 3-covered. (2) If r = 3g + h, b = 3h + g and the points in R ¿ B are in convex position, then at least r + b - 4 points can be K1, 3-covered, and this bound is tight. (3) There are arbitrarily large point sets R ¿ B in general position, with r = b + 1, such that at most r + b - 5 points can be K1, 3-covered. (4) If b = r = 3b, then at least 9 8 (r + b- 8) points of R ¿ B can be K1, 3-covered. For r > 3b, there are too many red points and at least r - 3b of them will remain uncovered in any K1, 3-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings
Tubular free by cyclic groups act freely on CAT(0) cube complexes
AbstractWe identify when a tubular group (the fundamental group of a ûnite graph of groups with ℤ2 vertex and ℤ edge groups) is free by cyclic and show, using Wise’s equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex.</jats:p
On Generalizations of Supereulerian Graphs
A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let and be the edge-connectivity and the minimum degree of a graph , respectively. For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . This dissertation is devoted to providing some results on -supereulerian graphs and supereulerian hypergraphs.
In Chapter 2, we determine the value of the smallest integer such that every -edge-connected graph is -supereulerian as follows:
j(s,t) = \left\{ \begin{array}{ll} \max\{4, t + 2\} & \mbox{ if $0 \le s \le 1$, or $(s,t) \in \{(2,0), (2,1), (3,0),(4,0)\}$,} \\ 5 & \mbox{ if $(s,t) \in \{(2,2), (3,1)\}$,} \\ s + t + \frac{1 - (-1)^s}{2} & \mbox{ if $s \ge 2$ and $s+t \ge 5$. } \end{array} \right.
As applications, we characterize -supereulerian graphs when in terms of edge-connectivities, and show that when , -supereulerianicity is polynomially determinable.
In Chapter 3, for a subset with , a necessary and sufficient condition for to be a contractible configuration for supereulerianicity is obtained. We also characterize the -supereulerianicity of when . These results are applied to show that if is -supereulerian with , then for any permutation on the vertex set , the permutation graph is -supereulerian if and only if .
For a non-negative integer , a graph is -Hamiltonian if the removal of any vertices results in a Hamiltonian graph. Let and denote the smallest integer such that the iterated line graph is -supereulerian and -Hamiltonian, respectively. In Chapter 4, for a simple graph , we establish upper bounds for and . Specifically, the upper bound for the -Hamiltonian index sharpens the result obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].
Harary and Nash-Williams in 1968 proved that the line graph of a graph is Hamiltonian if and only if has a dominating closed trail, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint spanning trees is a contractible configuration for supereulerianicity. In Chapter 5, utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Kir\\u27aly and Kriesell in 2003, we generalize the above-mentioned results of Harary and Nash-Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are Hamiltonian, and showing that every 2-partition-connected hypergraph is a contractible configuration for supereulerianicity.
Applying the adjacency matrix of a hypergraph defined by Rodr\\u27iguez in 2002, let be the second largest adjacency eigenvalue of . In Chapter 6, we prove that for an integer and a -uniform hypergraph of order with even, the minimum degree and , if , then is -edge-connected. %.
Some discussions are displayed in the last chapter. We extend the well-known Thomassen Conjecture that every 4-connected line graph is Hamiltonian to hypergraphs. The -supereulerianicity of hypergraphs is another interesting topic to be investigated in the future
The balanced 2-median and 2-maxian problems on a tree
This paper deals with the facility location problems with balancing on
allocation clients to servers. Two bi-objective models are considered, in which
one objective is the traditional p-median or p-maxian objective and the second
is to minimize the maximum demand volume allocated to any facility. An edge
deletion method with time complexity O(n^2) is presented for the balanced
-median problem on a tree. For the balanced 2-maxian problem, it is shown
the optimal solution is two end vertices of the diameter of the tree, which can
be obtained in a linear time.Comment: 19 page
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