29,158 research outputs found
Large Induced Forests in Planar Graphs
Tasandilised graafid on graafid, mida saab joonistada tasandile nii, et tema servad ei lõiku üksteisega mujal kui tippudes. Selles töös me uurime, kui suuri indutseeritud metsi on alati võimalik tasandilistes graafides leida. Praegu parim teadaolev tulemus on pärit Borodinilt, mille kohaselt igas tasandilises graafis leidub indutseeritud mets, mis sisaldab vähemalt 2/5 tema tippudest. Selles tööd anname me osalise tulemuse selle hinnangu parandamise suunas. Täpsemalt, me näitame, et minimaalne vastunäide meie parandatud tulemusele ei sisalda tippe, mille aste on väiksem kui 4, ja et selles sisalduvad astmega 4 tipud saavad olla vaid üht kindlat tüüpi.Planar graphs are graphs that can be drawn on a plane so that its edges do not cross each other (other that at their endpoints). In this thesis, we study the size of induced forests in planar graphs. The current best result by Borodin states that every planar graph has an induced forest that contains at least 2/5 of its vertices. In this thesis, we give partial results towards improving this bound. Specifically, we show that a minimal counterexample to an improved bound has minimal degree at least 3 and can contain only a specific type of vertices with degree 4
Equitable partition of graphs into induced forests
An equitable partition of a graph is a partition of the vertex-set of
such that the sizes of any two parts differ by at most one. We show that every
graph with an acyclic coloring with at most colors can be equitably
partitioned into induced forests. We also prove that for any integers
and , any -degenerate graph can be equitably
partitioned into induced forests.
Each of these results implies the existence of a constant such that for
any , any planar graph has an equitable partition into induced
forests. This was conjectured by Wu, Zhang, and Li in 2013.Comment: 4 pages, final versio
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 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
- …