8 research outputs found
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
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
Edge-partitioning regular graphs for ring traffic grooming with a priori placement od the ADMs
We study the following graph partitioning problem: Given two positive integers C
and Δ, find the least integer M(C,Δ) such that the edges of any graph with maximum degree at
most Δ can be partitioned into subgraphs with at most C edges and each vertex appears in at most
M(C,Δ) subgraphs. This problem is naturally motivated by traffic grooming, which is a major
issue in optical networks. Namely, we introduce a new pseudodynamic model of traffic grooming in
unidirectional rings, in which the aim is to design a network able to support any request graph with
a given bounded degree. We show that optimizing the equipment cost under this model is essentially
equivalent to determining the parameter M(C, Δ). We establish the value of M(C, Δ) for almost all
values of C and Δ, leaving open only the case where Δ ≥ 5 is odd, Δ (mod 2C) is between 3 and
C − 1, C ≥ 4, and the request graph does not contain a perfect matching. For these open cases, we
provide upper bounds that differ from the optimal value by at most one.Peer ReviewedPostprint (published version
Decomposing cubic graphs into isomorphic linear forests
A common problem in graph colouring seeks to decompose the edge set of a
given graph into few similar and simple subgraphs, under certain divisibility
conditions. In 1987 Wormald conjectured that the edges of every cubic graph on
vertices can be partitioned into two isomorphic linear forests. We prove
this conjecture for large connected cubic graphs. Our proof uses a wide range
of probabilistic tools in conjunction with intricate structural analysis, and
introduces a variety of local recolouring techniques.Comment: 49 pages, many figure
On linear k-arboricity
International audienc