Some remarks on Hajós’ conjecture

AbstractHajós’ conjecture is false for almost all graphs but only few explicit counterexamples have appeared in the literature. We relate Hajós’ conjecture to Ramsey theory, perfect graphs, and the maximum cut problem and obtain thereby new classes of explicit counterexamples. On the other hand, we show that some of the graphs which Catlin conjectured to be counterexamples to Hajós’ conjecture satisfy the conjecture, and we characterize completely the graphs which satisfy Catlin's conjecture

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

Graphs of polyhedra; polyhedra as graphs

AbstractRelations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and inconsistency that pervade much of the literature dealing with polyhedra more general than the convex ones

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Retrieval-, Distributed-, and Interleaved Practice in the Classroom:A Systematic Review

Three of the most effective learning strategies identified are retrieval practice, distributed practice, and interleaved practice, also referred to as desirable difficulties. However, it is yet unknown to what extent these three practices foster learning in primary and secondary education classrooms (as opposed to the laboratory and/or tertiary education classrooms, where most research is conducted) and whether these strategies affect different students differently. To address these gaps, we conducted a systematic review. Initial and detailed screening of 869 documents found in a threefold search resulted in a pool of 29 journal articles published from 2006 through June 2020. Seventy-five effect sizes nested in 47 experiments nested in 29 documents were included in the review. Retrieval- and interleaved practice appeared to benefit students’ learning outcomes quite consistently; distributed practice less so. Furthermore, only cognitive Student*Task characteristics (i.e., features of the student’s cognition regarding the task, such as initial success) appeared to be significant moderators. We conclude that future research further conceptualising and operationalising initial effort is required, as is a differentiated approach to implementing desirable difficulties