643 research outputs found
Rooted structures in graphs: a project on Hadwiger's conjecture, rooted minors, and Tutte cycles
Hadwigers Vermutung ist eine der anspruchsvollsten Vermutungen für Graphentheoretiker und bietet eine weitreichende Verallgemeinerung des Vierfarbensatzes. Ausgehend von dieser offenen Frage der strukturellen Graphentheorie werden gewurzelte Strukturen in Graphen diskutiert. Eine Transversale einer Partition ist definiert als eine Menge, welche genau ein Element aus jeder Menge der Partition enthält und sonst nichts. Für einen Graphen G und eine Teilmenge T seiner Knotenmenge ist ein gewurzelter Minor von G ein Minor, der T als Transversale seiner Taschen enthält. Sei T eine Transversale einer Färbung eines Graphen, sodass es ein System von kanten-disjunkten Wegen zwischen allen Knoten aus T gibt; dann stellt sich die Frage, ob es möglich ist, die Existenz eines vollständigen, in T gewurzelten Minors zu gewährleisten. Diese Frage ist eng mit Hadwigers Vermutung verwoben: Eine positive Antwort würde Hadwigers Vermutung für eindeutig färbbare Graphen bestätigen. In dieser Arbeit wird ebendiese Fragestellung untersucht sowie weitere Konzepte vorgestellt, welche bekannte Ideen der strukturellen Graphentheorie um eine Verwurzelung erweitern. Beispielsweise wird diskutiert, inwiefern hoch zusammenhängende Teilmengen der Knotenmenge einen hoch zusammenhängenden, gewurzelten Minor erzwingen. Zudem werden verschiedene Ideen von Hamiltonizität in planaren und nicht-planaren Graphen behandelt.Hadwiger's Conjecture is one of the most tantalising conjectures for graph theorists and offers a far-reaching generalisation of the Four-Colour-Theorem. Based on this major issue in structural graph theory, this thesis explores rooted structures in graphs. A transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. Given a graph G and a subset T of its vertex set, a rooted minor of G is a minor such that T is a transversal of its branch set. Assume that a graph has a transversal T of one of its colourings such that there is a system of edge-disjoint paths between all vertices from T; it comes natural to ask whether such graphs contain a minor rooted at T. This question of containment is strongly related to Hadwiger's Conjecture; indeed, a positive answer would prove Hadwiger's Conjecture for uniquely colourable graphs. This thesis studies the aforementioned question and besides, presents several other concepts of attaching rooted relatedness to ideas in structural graph theory. For instance, whether a highly connected subset of the vertex set forces a highly connected rooted minor. Moreover, several ideas of Hamiltonicity in planar and non-planar graphs are discussed
New bounds on domination and independence in graphs
We propose new bounds on the domination number and on the independence number
of a graph and show that our bounds compare favorably to recent ones. Our
bounds are obtained by using the Bhatia-Davis inequality linking the variance,
the expected value, the minimum, and the maximum of a random variable with
bounded distribution
On Selkow's bound on the independence number of graphs
For a graph G with vertex set V (G) and independence number α(G), Selkow [A Probabilistic lower bound on the independence number of graphs, Discrete Math. 132 (1994) 363–365] established the famous lower bound ∑v∈V(G)1d(v)+1(1+max{d(v)d(v)+1-∑u∈N(v)1d(u)+1,0}) on α (G), where N(v) and d(v) = |N(v)| denote the neighborhood and the degree of a vertex v ∈ V (G), respectively. However, Selkow’s original proof of this result is incorrect. We give a new probabilistic proof of Selkow’s bound here
Kempe Chains and Rooted Minors
A (minimal) transversal of a partition is a set which contains exactly one
element from each member of the partition and nothing else. A coloring of a
graph is a partition of its vertex set into anticliques, that is, sets of
pairwise nonadjacent vertices. We study the following problem: Given a
transversal of a proper coloring of some graph , is there
a partition of a subset of into connected sets such that
is a transversal of and such that two sets of
are adjacent if their corresponding vertices from are connected by a path
in using only two colors?
It has been conjectured by the first author that for any transversal of a
coloring of order of some graph such that any pair of
color classes induces a connected graph, there exists such a partition
with pairwise adjacent sets (which would prove Hadwiger's
conjecture for the class of uniquely optimally colorable graphs); this is open
for each , here we give a proof for the case that and the
subgraph induced by is connected. Moreover, we show that for , it
is not sufficient for the existence of as above just to force
any two transversal vertices to be connected by a 2-colored path
Strong modeling limits of graphs with bounded tree-width
The notion of first order convergence of graphs unifies the notions of
convergence for sparse and dense graphs. Ne\v{s}et\v{r}il and Ossona de Mendez
[J. Symbolic Logic 84 (2019), 452--472] proved that every first order
convergent sequence of graphs from a nowhere-dense class of graphs has a
modeling limit and conjectured the existence of such modeling limits with an
additional property, the strong finitary mass transport principle. The
existence of modeling limits satisfying the strong finitary mass transport
principle was proved for first order convergent sequences of trees by
Ne\v{s}et\v{r}il and Ossona de Mendez [Electron. J. Combin. 23 (2016), P2.52]
and for first order sequences of graphs with bounded path-width by Gajarsk\'y
et al. [Random Structures Algorithms 50 (2017), 612--635]. We establish the
existence of modeling limits satisfying the strong finitary mass transport
principle for first order convergent sequences of graphs with bounded
tree-width.Comment: arXiv admin note: text overlap with arXiv:1504.0812
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