38 research outputs found
Interleaved adjoints on directed graphs
For an integer k >= 1, the k-th interlacing adjoint of a digraph G is the
digraph i_k(G) with vertex-set V(G)^k, and arcs ((u_1, ..., u_k), (v_1, ...,
v_k)) such that (u_i,v_i) \in A(G) for i = 1, ..., k and (v_i, u_{i+1}) \in
A(G) for i = 1, ..., k-1. For every k we derive upper and lower bounds for the
chromatic number of i_k(G) in terms of that of G. In particular, we find tight
bounds on the chromatic number of interlacing adjoints of transitive
tournaments. We use this result in conjunction with categorial properties of
adjoint functors to derive the following consequence. For every integer ell,
there exists a directed path Q_{\ell} of algebraic length ell which admits
homomorphisms into every directed graph of chromatic number at least 4. We
discuss a possible impact of this approach on the multifactor version of the
weak Hedetniemi conjecture
Orientations and -colourings of graphs
summary:We provide the list of all paths with at most arcs with the property that if a graph admits an orientation such that one of the paths in our list admits no homomorphism to , then is -colourable
On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling
Vertex colouring is a well-known problem in combinatorial optimisation, whose
alternative integer programming formulations have recently attracted
considerable attention. This paper briefly surveys seven known formulations of
vertex colouring and introduces a formulation of vertex colouring using a
suitable clique partition of the graph. This formulation is applicable in
timetabling applications, where such a clique partition of the conflict graph
is given implicitly. In contrast with some alternatives, the presented
formulation can also be easily extended to accommodate complex performance
indicators (``soft constraints'') imposed in a number of real-life course
timetabling applications. Its performance depends on the quality of the clique
partition, but encouraging empirical results for the Udine Course Timetabling
problem are reported
Homomorphisms and Structural Properties of Relational Systems
Two main topics are considered: The characterisation of finite homomorphism
dualities for relational structures, and the splitting property of maximal
antichains in the homomorphism order.Comment: PhD Thesis, 77 pages, 14 figure
Collection of abstracts of the 24th European Workshop on Computational Geometry
International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop
Ahlfors circle maps and total reality: from Riemann to Rohlin
This is a prejudiced survey on the Ahlfors (extremal) function and the weaker
{\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e.
those (branched) maps effecting the conformal representation upon the disc of a
{\it compact bordered Riemann surface}. The theory in question has some
well-known intersection with real algebraic geometry, especially Klein's
ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a
gallery of pictures quite pleasant to visit of which we have attempted to trace
the simplest representatives. This drifted us toward some electrodynamic
motions along real circuits of dividing curves perhaps reminiscent of Kepler's
planetary motions along ellipses. The ultimate origin of circle maps is of
course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass.
Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found
in Klein (what we failed to assess on printed evidence), the pivotal
contribution belongs to Ahlfors 1950 supplying an existence-proof of circle
maps, as well as an analysis of an allied function-theoretic extremal problem.
Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree
controls than available in Ahlfors' era. Accordingly, our partisan belief is
that much remains to be clarified regarding the foundation and optimal control
of Ahlfors circle maps. The game of sharp estimation may look narrow-minded
"Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to
contemplate how conformal and algebraic geometry are fighting together for the
soul of Riemann surfaces. A second part explores the connection with Hilbert's
16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by
including now Rohlin's theory (v.2
Quasi-Modal Encounters Of The Third Kind: The Filling-In Of Visual Detail
Although Pessoa et al. imply that many aspects of the filling-in debate may be displaced by a regard for active vision, they remain loyal to naive neural reductionist explanations of certain pieces of psychophysical evidence. Alternative interpretations are provided for two specific examples and a new category of filling-in (of visual detail) is proposed