900 research outputs found
On the facial Thue choice index via entropy compression
A sequence is nonrepetitive if it contains no identical consecutive
subsequences. An edge colouring of a path is nonrepetitive if the sequence of
colours of its consecutive edges is nonrepetitive. By the celebrated
construction of Thue, it is possible to generate nonrepetitive edge colourings
for arbitrarily long paths using only three colours. A recent generalization of
this concept implies that we may obtain such colourings even if we are forced
to choose edge colours from any sequence of lists of size 4 (while sufficiency
of lists of size 3 remains an open problem). As an extension of these basic
ideas, Havet, Jendrol', Sot\'ak and \v{S}krabul'\'akov\'a proved that for each
plane graph, 8 colours are sufficient to provide an edge colouring so that
every facial path is nonrepetitively coloured. In this paper we prove that the
same is possible from lists, provided that these have size at least 12. We thus
improve the previous bound of 291 (proved by means of the Lov\'asz Local
Lemma). Our approach is based on the Moser-Tardos entropy-compression method
and its recent extensions by Grytczuk, Kozik and Micek, and by Dujmovi\'c,
Joret, Kozik and Wood
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
Characterizing Circular Colouring Mixing for
Given a graph , the -mixing problem asks: Can one obtain all
-colourings of , starting from one -colouring , by changing the
colour of only one vertex at a time, while at each step maintaining a
-colouring? More generally, for a graph , the -mixing problem asks:
Can one obtain all homomorphisms , starting from one homomorphism ,
by changing the image of only one vertex at a time, while at each step
maintaining a homomorphism ?
This paper focuses on a generalization of -colourings, namely
-circular colourings. We show that when , a graph
is -mixing if and only if for any -colouring of , and
any cycle of , the wind of the cycle under the colouring equals a
particular value (which intuitively corresponds to having no wind). As a
consequence we show that -mixing is closed under a restricted
homomorphism called a fold. Using this, we deduce that -mixing is
co-NP-complete for all , and by similar ideas we show that if
the circular chromatic number of a connected graph is ,
then folds to . We use the characterization to settle a
conjecture of Brewster and Noel, specifically that the circular mixing number
of bipartite graphs is . Lastly, we give a polynomial time algorithm for
-mixing in planar graphs when .Comment: 21 page
Logarithmic Weisfeiler-Leman Identifies All Planar Graphs
The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of iterations needed to obtain the final output is crucial for the parallelisability of the algorithm.
We show that there is a constant k such that every planar graph can be identified (that is, distinguished from every non-isomorphic graph) by the k-dimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3-connected planar graphs.
The number of iterations needed by the k-dimensional WL algorithm to identify a graph corresponds to the quantifier depth of a sentence that defines the graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth
A plane version of the Fleischner theorem
Let be a family of all -regular -connected plane multigraphs
without loops. We prove the following plane version of the Fleischner theorem:
Let be a graph in . For every -factor of having
-components there exists a plane graph having a Hamilton cycle omitting
all edges of and such that ,
and . Moreover, if is
simple, then is simple too.Comment: 5 pages, 2 figure
A Comprehensive Performance Evaluation of Deformable Face Tracking "In-the-Wild"
Recently, technologies such as face detection, facial landmark localisation
and face recognition and verification have matured enough to provide effective
and efficient solutions for imagery captured under arbitrary conditions
(referred to as "in-the-wild"). This is partially attributed to the fact that
comprehensive "in-the-wild" benchmarks have been developed for face detection,
landmark localisation and recognition/verification. A very important technology
that has not been thoroughly evaluated yet is deformable face tracking
"in-the-wild". Until now, the performance has mainly been assessed
qualitatively by visually assessing the result of a deformable face tracking
technology on short videos. In this paper, we perform the first, to the best of
our knowledge, thorough evaluation of state-of-the-art deformable face tracking
pipelines using the recently introduced 300VW benchmark. We evaluate many
different architectures focusing mainly on the task of on-line deformable face
tracking. In particular, we compare the following general strategies: (a)
generic face detection plus generic facial landmark localisation, (b) generic
model free tracking plus generic facial landmark localisation, as well as (c)
hybrid approaches using state-of-the-art face detection, model free tracking
and facial landmark localisation technologies. Our evaluation reveals future
avenues for further research on the topic.Comment: E. Antonakos and P. Snape contributed equally and have joint second
authorshi
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