14 research outputs found
Internal Partitions of Regular Graphs
An internal partition of an -vertex graph is a partition of
such that every vertex has at least as many neighbors in its own part as in the
other part. It has been conjectured that every -regular graph with
vertices has an internal partition. Here we prove this for . The case
is of particular interest and leads to interesting new open problems on
cubic graphs. We also provide new lower bounds on and find new families
of graphs with no internal partitions. Weighted versions of these problems are
considered as well
A note on 2--bisections of claw--free cubic graphs
A \emph{--bisection} of a bridgeless cubic graph is a --colouring
of its vertex set such that the colour classes have the same cardinality and
all connected components in the two subgraphs induced by the colour classes
have order at most . Ban and Linial conjectured that {\em every bridgeless
cubic graph admits a --bisection except for the Petersen graph}.
In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs
Flows and bisections in cubic graphs
A -weak bisection of a cubic graph is a partition of the vertex-set of
into two parts and of equal size, such that each connected
component of the subgraph of induced by () is a tree of at
most vertices. This notion can be viewed as a relaxed version of
nowhere-zero flows, as it directly follows from old results of Jaeger that
every cubic graph with a circular nowhere-zero -flow has a -weak bisection. In this paper we study problems related to the
existence of -weak bisections. We believe that every cubic graph which has a
perfect matching, other than the Petersen graph, admits a 4-weak bisection and
we present a family of cubic graphs with no perfect matching which do not admit
such a bisection. The main result of this article is that every cubic graph
admits a 5-weak bisection. When restricted to bridgeless graphs, that result
would be a consequence of the assertion of the 5-flow Conjecture and as such it
can be considered a (very small) step toward proving that assertion. However,
the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio
On 2-bisections and monochromatic edges in claw-free cubic multigraphs
A -bisection of a multigraph is a partition of its vertex set into two
parts of the same cardinality such that every component of each part has at
most vertices. Cui and Liu shown that every claw-free cubic multigraph
contains a -bisection, while Eom and Ozeki constructed specific
-bisections with bounded number of monochromatic edges. Their bound is the
best possible for claw-free cubic simple graphs. In this note, we extend the
latter result to the larger family of claw-free cubic multigraphsComment: 6 pages, 3 figure
Colourings of cubic graphs inducing isomorphic monochromatic subgraphs
A -bisection of a bridgeless cubic graph is a -colouring of its
vertex set such that the colour classes have the same cardinality and all
connected components in the two subgraphs induced by the colour classes
(monochromatic components in what follows) have order at most . Ban and
Linial conjectured that every bridgeless cubic graph admits a -bisection
except for the Petersen graph. A similar problem for the edge set of cubic
graphs has been studied: Wormald conjectured that every cubic graph with
has a -edge colouring such that the two
monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose
components are paths). Finally, Ando conjectured that every cubic graph admits
a bisection such that the two induced monochromatic subgraphs are isomorphic.
In this paper, we give a detailed insight into the conjectures of Ban-Linial
and Wormald and provide evidence of a strong relation of both of them with
Ando's conjecture. Furthermore, we also give computational and theoretical
evidence in their support. As a result, we pose some open problems stronger
than the above mentioned conjectures. Moreover, we prove Ban-Linial's
conjecture for cubic cycle permutation graphs.
As a by-product of studying -edge colourings of cubic graphs having linear
forests as monochromatic components, we also give a negative answer to a
problem posed by Jackson and Wormald about certain decompositions of cubic
graphs into linear forests.Comment: 33 pages; submitted for publicatio
Ban--Linial's Conjecture and treelike snarks
A bridgeless cubic graph is said to have a 2-bisection if there exists a
2-vertex-colouring of (not necessarily proper) such that: (i) the colour
classes have the same cardinality, and (ii) the monochromatic components are
either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that
every bridgeless cubic graph, apart from the well-known Petersen graph, admits
a 2-bisection. In the same paper it was shown that every Class I bridgeless
cubic graph admits such a bisection. The Class II bridgeless cubic graphs which
are critical to many conjectures in graph theory are snarks, in particular,
those with excessive index at least 5, that is, whose edge-set cannot be
covered by four perfect matchings. Moreover, Esperet et al. state that a
possible counterexample to Ban--Linial's Conjecture must have circular flow
number at least 5. The same authors also state that although empirical evidence
shows that several graphs obtained from the Petersen graph admit a 2-bisection,
they can offer nothing in the direction of a general proof. Despite some
sporadic computational results, until now, no general result about snarks
having excessive index and circular flow number both at least 5 has been
proven. In this work we show that treelike snarks, which are an infinite family
of snarks heavily depending on the Petersen graph and with both their circular
flow number and excessive index at least 5, admit a 2-bisection.Comment: 10 pages, 6 figure
Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization
A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [IPEC 2018], we introduce a natural generalization of this problem, which we call d-Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d-Cut on (2d+2)-regular graphs and a polynomial algorithm for graphs of maximum degree at most d+2. The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [IPEC 2018], is a polynomial kernel for d-Cut for every positive integer d, parameterized by the distance to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time O^*(2^n)
Isomorphic bisections of cubic graphs
Graph partitioning, or the dividing of a graph into two or more parts based on certain conditions, arises naturally throughout discrete mathematics, and problems of this kind have been studied extensively. In the 1990s, Ando conjectured that the vertices of every cubic graph can be partitioned into two parts that induce isomorphic subgraphs. Using probabilistic methods together with delicate recolouring arguments, we prove Ando's conjecture for large connected graphs