31,560 research outputs found
Packing Directed Cycles Quarter- and Half-Integrally
The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph
that does not admit a family of vertex-disjoint cycles contains a feedback
vertex set (a set of vertices hitting all cycles in the graph) of size . After being known for long as Younger's conjecture, a similar
statement for directed graphs has been proven in 1996 by Reed, Robertson,
Seymour, and Thomas. However, in their proof, the dependency of the size of the
feedback vertex set on the size of vertex-disjoint cycle packing is not
elementary.
We show that if we compare the size of a minimum feedback vertex set in a
directed graph with the quarter-integral cycle packing number, we obtain a
polynomial bound. More precisely, we show that if in a directed graph there
is no family of cycles such that every vertex of is in at most four of
the cycles, then there exists a feedback vertex set in of size .
Furthermore, a variant of our proof shows that if in a directed graph there
is no family of cycles such that every vertex of is in at most two of
the cycles, then there exists a feedback vertex set in of size .
On the way there we prove a more general result about quarter-integral
packing of subgraphs of high directed treewidth: for every pair of positive
integers and , if a directed graph has directed treewidth
, then one can find in a family of
subgraphs, each of directed treewidth at least , such that every vertex of
is in at most four subgraphs.Comment: Accepted to European Symposium on Algorithms (ESA '19
The Element Spectrum Of A Graph
Characterizations of graphs and matroids that have cycles or circuits of specified cardinality have been given by authors including Edmonds, Junior, Lemos, Murty, Reid, Young, and Wu. In particular, a matroid with circuits of a single cardinality is called a Matroid Design. We consider a generalization of this problem by assigning a weight function to the edges of a graph. We characterize when it is possible to assign a positive integer value weight function to a simple 3-connected graph G such that the graph G contains an edge that is only in cycles of two different weights. For example, as part of the main theorem we show that if this assignment is possible, then the graph G is an extension of a three-wheel, a four-wheel, a five-wheel , K3,n, a prism, a certain seven-vertex graph, or a certain eight-vertex graph, or G is obtained from the latter three graphs by attaching triads in a certain manner. The reason for assigning weights is that if each edge of such a graph is subdivided according to the weight function, then the resulting subdivided graph will contain cycles through a fixed edge of just a few different cardinalities. We consider the case where the graph has a pair of vertex-disjoint cycles and the case where the graph does not have a pair of vertex-disjoint cycles. Results from graph structure theory are used to give these characterizations
Partitioning 2-edge-colored graphs by monochromatic paths and cycles
We present results on partitioning the vertices of -edge-colored graphs
into monochromatic paths and cycles. We prove asymptotically the two-color case
of a conjecture of S\'ark\"ozy: the vertex set of every -edge-colored graph
can be partitioned into at most monochromatic cycles, where
denotes the independence number of . Another direction, emerged
recently from a conjecture of Schelp, is to consider colorings of graphs with
given minimum degree. We prove that apart from vertices, the vertex
set of any -edge-colored graph with minimum degree at least
(1+\eps){3|V(G)|\over 4} can be covered by the vertices of two vertex
disjoint monochromatic cycles of distinct colors. Finally, under the assumption
that does not contain a fixed bipartite graph , we show that
in every -edge-coloring of , vertices can be covered by two
vertex disjoint paths of different colors, where is a constant depending
only on . In particular, we prove that , which is best possible
Minimal Spectrum-Sums of Bipartite Graphs with Exactly Two Vertex-Disjoint Cycles
The spectrum-sum of a graph is defined as the sum of the absolute values of its eigenvalues.
The graphs with minimal spectrum-sums in the class of connected bipartite graphs with exactly
two vertex-disjoint cycles, in the class of connected bipartite graphs with exactly two vertex-
-disjoint cycles whose lengths are congruent with 2 modulo 4, and in the class of connected bipartite
graphs with exactly two vertex-disjoint cycles one of which has length congruent with 2
modulo 4, are determined, respectively
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