Proof of a local antimagic conjecture

An antimagic labelling of a graph $G$ is a bijection $f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \& Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \& Lyngsie (2017)) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than $K_2$ admits a local antimagic labelling. We prove this latter conjecture using the probabilistic method. Thus the parameter of local antimagic chromatic number, introduced by Arumugam et al., is well-defined for every connected graph other than $K_2$ .Comment: Final version for publication in DMTCS. Changes from previous version are formatting to journal style and correction of two minor typographical error

A QBF-based Formalization of Abstract Argumentation Semantics

Supported by the National Research Fund, Luxembourg (LAAMI project) and by the Engineering and Physical Sciences Research Council (EPSRC, UK), grant ref. EP/J012084/1 (SAsSY project).Peer reviewedPostprin

Knotenfärbungen mit Abstandsbedingungen

Knotenfärbungen mit Abstandsbedingungen sind graphentheoretische Konzepte, motiviert durch das praktische Problem der Frequenzzuweisung in Mobilfunknetzen. In der Arbeit werden verschiedene Varianten solcher Färbungen vorgestellt. Für (Listen-)Färbungen mit einer beliebigen Anzahl r von Abstandsbedingungen werden allgemeine Eigenschaften und Schranken für die benötigte Anzahl von Farben bewiesen. Anschließend wird der Spezialfall r=2 behandelt. Färbungen mit zwei Abstandsbedingungen - die sogenannten L(d,s)-Labellings - werden für eine Reihe von Graphenklassen untersucht, u.a. für reguläre Parkettierungen, Weg- und Kreispotenzen und Graphen mit Durchmesser 2. Die Listenversion dieser Färbungen - die sogenannten L(d,s)-List Labellings - werden für Wege, Sterne, Kreise und Kakteen betrachtet. Ferner werden Untersuchungen zum Zusammenhang von L(2,1)-Labellings und L(2,1)-List Labellings bei speziellen Bäumen durchgeführt

The classification of Wada-type representations of braid groups

We give a classification of Wada-type representations of the braid groups, and solutions of a variant of the set-theoretical Yang-Baxter equation adapted to the free-product group structure. As a consequence, we prove Wada's conjecture: There are only seven types of Wada-type representations up to certain symmetries.Comment: 13 pages, 2 figures: Added Lemma 2.2, which was implicit in the previous version without proof and more explanation

Khovanov homotopy type, periodic links and localizations

Given an $m$-periodic link $L\subset S^3$, we show that the Khovanov spectrum \X_L constructed by Lipshitz and Sarkar admits a homology group action. We relate the Borel cohomology of \X_L to the equivariant Khovanov homology of $L$ constructed by the second author. The action of Steenrod algebra on the cohomology of \X_L gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying Dwyer-Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.Comment: 80 pages, 23 figures. The paper underwent a major revision. New version contains the proof of the Geometric Fixed Point Theorem (Theorem 1.4) for Khovanov homotopy type and annular Khovanov homotopy typ