1,470 research outputs found
On bipartite graphs of defect at most 4
We consider the bipartite version of the degree/diameter problem, namely,
given natural numbers {\Delta} \geq 2 and D \geq 2, find the maximum number
Nb({\Delta},D) of vertices in a bipartite graph of maximum degree {\Delta} and
diameter D. In this context, the Moore bipartite bound Mb({\Delta},D)
represents an upper bound for Nb({\Delta},D). Bipartite graphs of maximum
degree {\Delta}, diameter D and order Mb({\Delta},D), called Moore bipartite
graphs, have turned out to be very rare. Therefore, it is very interesting to
investigate bipartite graphs of maximum degree {\Delta} \geq 2, diameter D \geq
2 and order Mb({\Delta},D) - \epsilon with small \epsilon > 0, that is,
bipartite ({\Delta},D,-\epsilon)-graphs. The parameter \epsilon is called the
defect. This paper considers bipartite graphs of defect at most 4, and presents
all the known such graphs. Bipartite graphs of defect 2 have been studied in
the past; if {\Delta} \geq 3 and D \geq 3, they may only exist for D = 3.
However, when \epsilon > 2 bipartite ({\Delta},D,-\epsilon)-graphs represent a
wide unexplored area. The main results of the paper include several necessary
conditions for the existence of bipartite -graphs; the complete
catalogue of bipartite (3,D,-\epsilon)-graphs with D \geq 2 and 0 \leq \epsilon
\leq 4; the complete catalogue of bipartite ({\Delta},D,-\epsilon)-graphs with
{\Delta} \geq 2, 5 \leq D \leq 187 (D /= 6) and 0 \leq \epsilon \leq 4; and a
non-existence proof of all bipartite ({\Delta},D,-4)-graphs with {\Delta} \geq
3 and odd D \geq 7. Finally, we conjecture that there are no bipartite graphs
of defect 4 for {\Delta} \geq 3 and D \geq 5, and comment on some implications
of our results for upper bounds of Nb({\Delta},D).Comment: 25 pages, 14 Postscript figure
Towards Effective Exact Algorithms for the Maximum Balanced Biclique Problem
The Maximum Balanced Biclique Problem (MBBP) is a prominent model with
numerous applications. Yet, the problem is NP-hard and thus computationally
challenging. We propose novel ideas for designing effective exact algorithms
for MBBP. Firstly, we introduce an Upper Bound Propagation procedure to
pre-compute an upper bound involving each vertex. Then we extend an existing
branch-and-bound algorithm by integrating the pre-computed upper bounds. We
also present a set of new valid inequalities induced from the upper bounds to
tighten an existing mathematical formulation for MBBP. Lastly, we investigate
another exact algorithm scheme which enumerates a subset of balanced bicliques
based on our upper bounds. Experiments show that compared to existing
approaches, the proposed algorithms and formulations are more efficient in
solving a set of random graphs and large real-life instances
Topological lower bounds for the chromatic number: A hierarchy
This paper is a study of ``topological'' lower bounds for the chromatic
number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978,
in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology.
This conjecture stated that the \emph{Kneser graph} \KG_{m,n}, the graph with
all -element subsets of as vertices and all pairs of
disjoint sets as edges, has chromatic number . Several other proofs
have since been published (by B\'ar\'any, Schrijver, Dolnikov, Sarkaria, Kriz,
Greene, and others), all of them based on some version of the Borsuk--Ulam
theorem, but otherwise quite different. Each can be extended to yield some
lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe
that \emph{every} finite graph may be represented as a generalized Kneser
graph, to which the above bounds apply.)
We show that these bounds are almost linearly ordered by strength, the
strongest one being essentially Lov\'asz' original bound in terms of a
neighborhood complex. We also present and compare various definitions of a
\emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lov\'asz
and of \kriz). A suitable box complex is equivalent to Lov\'asz' complex, but
the construction is simpler and functorial, mapping graphs with homomorphisms
to -spaces with -maps.Comment: 16 pages, 1 figure. Jahresbericht der DMV, to appea
Defective and Clustered Choosability of Sparse Graphs
An (improper) graph colouring has "defect" if each monochromatic subgraph
has maximum degree at most , and has "clustering" if each monochromatic
component has at most vertices. This paper studies defective and clustered
list-colourings for graphs with given maximum average degree. We prove that
every graph with maximum average degree less than is
-choosable with defect . This improves upon a similar result by Havet and
Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with
maximum average degree , no bound on the number of colours
was previously known. The above result with solves this problem. It
implies that every graph with maximum average degree is
-choosable with clustering 2. This extends a
result of Kopreski and Yu [Discrete Math., 2017] to the setting of
choosability. We then prove two results about clustered choosability that
explore the trade-off between the number of colours and the clustering. In
particular, we prove that every graph with maximum average degree is
-choosable with clustering , and is
-choosable with clustering . As an
example, the later result implies that every biplanar graph is 8-choosable with
bounded clustering. This is the best known result for the clustered version of
the earth-moon problem. The results extend to the setting where we only
consider the maximum average degree of subgraphs with at least some number of
vertices. Several applications are presented
Height representation of XOR-Ising loops via bipartite dimers
The XOR-Ising model on a graph consists of random spin configurations on
vertices of the graph obtained by taking the product at each vertex of the
spins of two independent Ising models. In this paper, we explicitly relate loop
configurations of the XOR-Ising model and those of a dimer model living on a
decorated, bipartite version of the Ising graph. This result is proved for
graphs embedded in compact surfaces of genus g.
Using this fact, we then prove that XOR-Ising loops have the same law as
level lines of the height function of this bipartite dimer model. At
criticality, the height function is known to converge weakly in distribution to
a Gaussian free field.
As a consequence, results of this paper shed a light on the occurrence of the
Gaussian free field in the XOR-Ising model. In particular, they prove a
discrete analogue of Wilson's conjecture, stating that the scaling limit of
XOR-Ising loops are "contour lines" of the Gaussian free field.Comment: 41 pages, 10 figure
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