Remarks on the existence of uniquely partitionable planar graphs

We consider the problem of the existence of uniquely partitionable planar graphs. We survey some recent results and we prove the nonexistence of uniquely (D1,D1)-partitionable planar graphs with respect to the property D1 "to be a forest"

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

Forced color classes, intersection graphs and the strong perfect graph conjecture

AbstractIn 1996, A. Sebő[11] raised the following two conjectures concerned with the famous Strong Perfect Graph Conjecture: (1) Suppose that a minimally imperfect graph G has a vertex p incident to 2ω(G)−2 determined edges and that its complement Ḡ has a vertex q incident to 2α(G)−2 determined edges. (An edge of G is called determined if an ω-clique of G contains both of its endpoints.) Then G is an odd hole or an odd antihole. (2) Let v0 be a vertex of a partitionable graph G. And suppose A,B to be ω-cliques of G so that v0∈A∩B. If every ω-clique K containing the vertex v0 is contained in A∪B, then G is an odd hole or an odd antihole. In this paper, we will prove (1) for a minimally imperfect graph G such that (p,q) is a determined edge of either G or Ḡ, and prove (2) for a minimally imperfect graph G such that Ḡ is C4-free and edges of Ḡ are all determined edges

Partitioning a graph into disjoint cliques and a triangle-free graph

A graph G=(V,E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques (i.e., G[A] is P_3-free) and B induces a triangle-free graph (i.e., G[B] is K_3-free). In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K_4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomassé, Trotignon and Vušković. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs

On the Complexity of Hilbert Refutations for Partition

Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a given set of integers is not partitionable. We provide an explicit construction of a minimum-degree certificate, and then demonstrate that the Partition problem is equivalent to the determinant of a carefully constructed matrix called the partition matrix. In particular, we show that the determinant of the partition matrix is a polynomial that factors into an iteration over all possible partitions of W.Comment: Final versio

Colorings of graphs, digraphs, and hypergraphs

Brooks' Theorem ist eines der bekanntesten Resultate über Graphenfärbungen: Sei G ein zusammenhängender Graph mit Maximalgrad d. Ist G kein vollständiger Graph, so lassen sich die Ecken von G so mit d Farben färben, dass zwei benachbarte Ecken unterschiedlich gefärbt sind. In der vorliegenden Arbeit liegt der Fokus auf Verallgemeinerungen von Brooks Theorem für Färbungen von Hypergraphen und gerichteten Graphen. Eine Färbung eines Hypergraphen ist eine Färbung der Ecken so, dass keine Kante monochromatisch ist. Auf Hypergraphen erweitert wurde der Satz von Brooks von R.P. Jones. Im ersten Teil der Dissertation werden Möglichkeiten aufgezeigt, das Resultat von Jones weiter zu verallgemeinern. Kernstück ist ein Zerlegungsresultat: Zu einem Hypergraphen H und einer Folge f=(f_1,…,f_p) von Funktionen, welche von V(H) in die natürlichen Zahlen abbilden, wird untersucht, ob es eine Zerlegung von H in induzierte Unterhypergraphen H_1,…,H_p derart gibt, dass jedes H_i strikt f_i-degeneriert ist. Dies bedeutet, dass jeder Unterhypergraph H_i' von H_i eine Ecke v enthält, deren Grad in H_i' kleiner als f_i(v) ist. Es wird bewiesen, dass die Bedingung f_1(v)+…+f_p(v) \geq d_H(v) für alle v fast immer ausreichend für die Existenz einer solchen Zerlegung ist und gezeigt, dass sich die Ausnahmefälle gut charakterisieren lassen. Durch geeignete Wahl der Funktion f lassen sich viele bekannte Resultate ableiten, was im dritten Kapitel erörtert wird. Danach werden zwei weitere Verallgemeinerungen des Satzes von Jones bewiesen: Ein Theorem zu DP-Färbungen von Hypergraphen und ein Resultat, welches die chromatische Zahl eines Hypergraphen mit dessen maximalem lokalen Kantenzusammenhang verbindet. Der zweite Teil untersucht Färbungen gerichteter Graphen. Eine azyklische Färbung eines gerichteten Graphen ist eine Färbung der Eckenmenge des gerichteten Graphen sodass es keine monochromatischen gerichteten Kreise gibt. Auf dieses Konzept lassen sich viele klassische Färbungsresultate übertragen. Dazu zählt auch Brooks Theorem, wie von Mohar bewiesen wurde. Im siebten Kapitel werden DP-Färbungen gerichteter Graphen untersucht. Insbesondere erfolgt der Transfer von Mohars Theorem auf DP-Färbungen. Das darauffolgende Kapitel befasst sich mit kritischen gerichteten Graphen. Insbesondere werden Konstruktionen für diese angegeben und die gerichtete Version des Satzes von Hajós bewiesen.Brooks‘ Theorem is one of the most known results in graph coloring theory: Let G be a connected graph with maximum degree d >2. If G is not a complete graph, then there is a coloring of the vertices of G with d colors such that no two adjacent vertices get the same color. Based on Brooks' result, various research topics in graph coloring arose. Also, it became evident that Brooks' Theorem could be transferred to many other coloring-concepts. The present thesis puts its focus especially on two of those concepts: hypergraphs and digraphs. A coloring of a hypergraph H is a coloring of its vertices such that no edge is monochromatic. Brooks' Theorem for hypergraphs was obtained by R.P. Jones. In the first part of this thesis, we present several ways how to further extend Jones' theorem. The key element is a partition result, to which the second chapter is devoted. Given a hypergraph H and a sequence f=(f_1,…,f_p) of functions, we examine if there is a partition of $H$ into induced subhypergraphs H_1,…,H_p such that each of the H_i is strictly f_i-degenerate. This means that in each non-empty subhypergraph H_i' of H_i there is a vertex v having degree d_{H_i'}(v

The algebra of set functions I: The product theorem and duality

AbstractWe give a comprehensive introduction to the algebra of set functions and its generating functions. This algebraic tool allows us to formulate and prove a product theorem for the enumeration of functions of many different kinds, in particular injective functions, surjective functions, matchings and colourings of the vertices of a hypergraph. Moreover, we develop a general duality theory for counting functions