7 research outputs found
GráfszĂnezĂ©sek Ă©s gráfok felbontásai = Colorings and decompositions of graphs
A nem-ismĂ©tlĹ‘ szĂnezĂ©seket a vĂ©letlen mĂłdszer alkalmazhatĂłsága miatt kezdtĂ©k el vizsgálni. FelsĹ‘ korlátot adtunk a szĂnek számára, amely a maximum fok Ă©s a favastagság lineáris fĂĽggvĂ©nye. Olyan szĂnezĂ©seket is vizsgáltunk, amelyek egy sĂkgráf oldalain nem-ismĂ©tlĹ‘k. SejtĂ©s volt, hogy vĂ©ges sok szĂn elĂ©g. Ezt bizonyĂtottuk 24 szĂnnel. A kromatikus számot Ă©s a metszĂ©si számot algoritmikusan nehĂ©z meghatározni. EzĂ©rt meglepĹ‘ Albertson egy friss sejtĂ©se, amely kapcsolatot állĂt fel közöttĂĽk: ha egy gráf kromatikus száma r, akkor metszĂ©si száma legalább annyi, mint a teljes r csĂşcsĂş gráfĂ©. BizonyĂtottuk a sejtĂ©st, ha r<3.57n, valamint ha 12<r<17. Ez utĂłbbi azĂ©rt Ă©rdekes, mert a teljes r csĂşcsĂş gráf metszĂ©si száma csak r<13 esetĂ©n ismert. A tĂ©makör legfontosabb nyitott kĂ©rdĂ©se a Hadwiger-sejtĂ©s, mely szerint minden r-kromatikus gráf tartalmazza a teljes r csĂşcsĂş gráfot minorkĂ©nt. Ennek általánosĂtásakĂ©nt fogalmazták meg a lista szĂnezĂ©si Hadwiger sejtĂ©st: ha egy gráf nem tartalmaz teljes r csĂşcsĂş gráfot minorkĂ©nt, akkor az r-lista szĂnezhetĹ‘. Megmutattuk, hogy ez a sejtĂ©s hamis. Legalább cr szĂnre szĂĽksĂ©gĂĽnk van bizonyos gráfokra, ahol c=4/3. Thomassennel vetettĂĽk fel azt a kĂ©rdĂ©st, hogy milyen feltĂ©tel garantálja, hogy G Ă©lei felbonthatĂłk egy adott T fa pĂ©ldányaira. Legyen Y az a fa, melynek fokszámsorozata (1,1,1,2,3). Megmutattuk, hogy minden 287-szeresen Ă©lösszefĂĽggĹ‘ fa felbomlik Y-okra, ha az Ă©lszám oszthatĂł 4-gyel. | Nonrepetitive colorings often use the probabilistic method. We gave an upper bound as a linear function of the maximum degree and the tree-width. We also investigated colorings, which are nonrepetitive on faces of plane graphs. As conjectured, a finite number of colors suffice. We proved it by 24 colors. The chromatic and crossing numbers are both difficult to compute. The recent Albertson's conjecture is a surprising relation between the two: if the chromatic number is r, then the crossing number is at least the crossing number of the complete graph on r vertices. We proved this claim, if r<3.57n, or 12<r<17. The latter is remarkable, since the crossing number of the complete graph is only known for r<13. The most important open question of the field is Hadwiger's conjecture: every r-chromatic graph contains as a minor the complete graph on r vertices. As a generalisation, the following is the list coloring Hadwiger conjecture: if a graph does not contain as a minor the complete graph on r vertices , then the graph is r-list colorable. We proved the falsity of this claim. In our examples, at least cr colors are necessary, where c=4/3. Decomposition of graphs is well-studied. Thomassen and I posed the question of a sufficient connectivity condition, which guaranties a T-decomposition. Let Y be the tree with degree sequence (1,1,1,2,3). We proved every 287-edge connected graph has a Y-decomposition, if the size is divisible by four
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
Recommended from our members
Discrete Geometry
A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry
Recommended from our members
Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session
Discrete Geometry and Convexity in Honour of Imre Bárány
This special volume is contributed by the speakers of the Discrete Geometry and
Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference
is to celebrate the 70th birthday and the scientific achievements of professor
Imre Bárány, a pioneering researcher of discrete and convex geometry, topological
methods, and combinatorics. The extended abstracts presented here are written by
prominent mathematicians whose work has special connections to that of professor
Bárány. Topics that are covered include: discrete and combinatorial geometry,
convex geometry and general convexity, topological and combinatorial methods.
The research papers are presented here in two sections. After this preface and a
short overview of Imre Bárány’s works, the main part consists of 20 short but very
high level surveys and/or original results (at least an extended abstract of them)
by the invited speakers. Then in the second part there are 13 short summaries of
further contributed talks.
We would like to dedicate this volume to Imre, our great teacher, inspiring
colleague, and warm-hearted friend