23 research outputs found
Clique immersions in graphs of independence number two with certain forbidden subgraphs
The Lescure-Meyniel conjecture is the analogue of Hadwiger's conjecture for
the immersion order. It states that every graph contains the complete graph
as an immersion, and like its minor-order counterpart it is open
even for graphs with independence number 2. We show that every graph with
independence number and no hole of length between and
satisfies this conjecture. In particular, every -free graph
with satisfies the Lescure-Meyniel conjecture. We give
another generalisation of this corollary, as follows. Let and be graphs
with independence number at most 2, such that . If is
-free, then satisfies the Lescure-Meyniel conjecture.Comment: 14 pages, 3 figures. The statements of lemmas 3.1, 4.1, and 4.2 are
slightly changed from the previous version in order to fix some minor errors
in the proofs of theorems 3.2 and 4.3. Shorter proof of Proposition 5.2 give
Clique immersion in graphs without fixed bipartite graph
A graph contains as an \emph{immersion} if there is an injective
mapping such that for each edge ,
there is a path in joining vertices and , and
all the paths , , are pairwise edge-disjoint. An analogue
of Hadwiger's conjecture for the clique immersions by Lescure and Meyniel
states that every graph contains as an immersion. We consider
the average degree condition and prove that for any bipartite graph , every
-free graph with average degree contains a clique immersion of order
, implying that Lescure and Meyniel's conjecture holds
asymptotically for graphs without fixed bipartite graph.Comment: 2 figure
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Improper colourings inspired by Hadwiger’s conjecture
Hadwiger’s Conjecture asserts that every Kt-minor-free graph has a proper (t − 1)-colouring. We relax the conclusion in Hadwiger’s Conjecture via improper colourings. We prove that every Kt-minor-free graph is (2t − 2)-colourable with monochromatic components of order at most 1/2 (t − 2). This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt-minor-free graph is (t − 1)-colourable with monochromatic degree at most t − 2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t-minorfree graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt-immersion are 2-colourable with bounded monochromatic degree
Immersion and clustered coloring
Hadwiger and Haj\'{o}s conjectured that for every positive integer ,
-minor free graphs and -topological minor free graphs are
properly -colorable, respectively. Clustered coloring version of these two
conjectures which only require monochromatic components to have bounded size
has been extensively studied. In this paper we consider the clustered coloring
version of the immersion-variant of Hadwiger's and Haj\'{o}s' conjecture
proposed by Lescure and Meyniel and independently by Abu-Khzam and Langston. We
determine the minimum number of required colors for -immersion free graphs,
for any fixed graph , up to a small additive absolute constant. Our result
is tight for infinitely many graphs .
A key machinery developed in this paper is a lemma that reduces a clustering
coloring problem on graphs to the one on the torsos of their tree-cut
decomposition or tree-decomposition. A byproduct of this machinery is a unified
proof of a result of Alon, Ding, Oporowski and Vertigan and a result of the
author and Oum about clustered coloring graphs of bounded maximum degree in
minor-closed families
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