190 research outputs found
Proof of a local antimagic conjecture
An antimagic labelling of a graph is a bijection
such that the sums
distinguish all vertices. A well-known conjecture of Hartsfield and Ringel
(1994) is that every connected graph other than 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 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 .Comment: Final version for publication in DMTCS. Changes from previous version
are formatting to journal style and correction of two minor typographical
error
Preferential attachment with choice
We consider the degree distributions of preferential attachment random graph
models with choice similar to those considered in recent work by Malyshkin and
Paquette and Krapivsky and Redner. In these models a new vertex chooses
vertices according to a preferential rule and connects to the vertex in the
selection with the th highest degree. For meek choice, where , we show
that both double exponential decay of the degree distribution and
condensation-like behaviour are possible, and provide a criterion to
distinguish between them. For greedy choice, where , we confirm that the
degree distribution asympotically follows a power law with logarithmic
correction when and shows condensation-like behaviour when .Comment: 17 pages, 1 figure. Accepted for publication in Random Structures and
Algorithm
Countable graphs are majority 3-choosable
The Unfriendly Partition Conjecture posits that every countable graph admits
a 2-colouring in which for each vertex there are at least as many bichromatic
edges containing that vertex as monochromatic ones. This is not known in
general, but it is known that a 3-colouring with this property always exists.
Anholcer, Bosek and Grytczuk recently gave a list-colouring version of this
conjecture, and proved that such a colouring exists for lists of size 4. We
improve their result to lists of size 3; the proof extends to directed acyclic
graphs. We also discuss some generalisations.Comment: 6 pages. Minor changes including adding a referenc
Determining triangulations and quadrangulations by boundary distances
We show that if a disc triangulation has all internal vertex degrees at least
6, then the full triangulation may be determined from the pairwise graph
distance between boundary vertices. A similar result holds for quadrangulations
with all internal degrees at least 4. This confirms a conjecture of Itai
Benjamini. Both degree bounds are best possible, and correspond to local
non-positive curvature. However, we show that a natural conjecture for a
"mixed" version of the two results is not true.Comment: 18 pages, 5 figure
The path minimises the average size of a connected induced subgraph
We prove that among all graphs of order n, the path uniquely minimises the
average order of its connected induced subgraphs. This confirms a conjecture of
Kroeker, Mol and Oellermann, and generalises a classical result of Jamison for
trees, as well as giving a new, shorter proof of the latter. While this paper
was being prepared, a different proof was given by Andrew Vince.Comment: 9 pages, 1 figure. Changed title, new figure and minor rewritin
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