6,130 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
Size of the Largest Induced Forest in Subcubic Graphs of Girth at least Four and Five
In this paper, we address the maximum number of vertices of induced forests
in subcubic graphs with girth at least four or five. We provide a unified
approach to prove that every 2-connected subcubic graph on vertices and
edges with girth at least four or five, respectively, has an induced forest on
at least or vertices, respectively, except
for finitely many exceptional graphs. Our results improve a result of Liu and
Zhao and are tight in the sense that the bounds are attained by infinitely many
2-connected graphs. Equivalently, we prove that such graphs admit feedback
vertex sets with size at most or , respectively.
Those exceptional graphs will be explicitly constructed, and our result can be
easily modified to drop the 2-connectivity requirement
Some Results on incidence coloring, star arboricity and domination number
Two inequalities bridging the three isolated graph invariants, incidence
chromatic number, star arboricity and domination number, were established.
Consequently, we deduced an upper bound and a lower bound of the incidence
chromatic number for all graphs. Using these bounds, we further reduced the
upper bound of the incidence chromatic number of planar graphs and showed that
cubic graphs with orders not divisible by four are not 4-incidence colorable.
The incidence chromatic numbers of Cartesian product, join and union of graphs
were also determined.Comment: 8 page
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