9 research outputs found
An ILP-based Proof System for the Crossing Number Problem
Formally, approaches based on mathematical programming are able to find provably optimal solutions.
However, the demands on a verifiable formal proof are typically much higher than the guarantees
we can sensibly attribute to implementations of mathematical programs. We consider this in the context of the crossing number problem, one of the most prominent problems in topological graph theory. The problem asks for the minimum number of edge crossings in any drawing of a given graph. Graph-theoretic proofs for this problem are known to be notoriously hard to obtain. At the same time, proofs even for very specific graphs are often of interest in crossing number research, as they can, e.g., form the basis for inductive proofs.
We propose a system to automatically generate a formal proof based on an ILP computation. Such a proof is (relatively) easily verifiable, and does not require the understanding of any complex ILP codes. As such, we hope our proof system may serve as a showcase for the necessary steps and central design goals of how to establish formal proof systems based on mathematical programming formulations
On the Crossing Number of the Cartesian Product of a Sunlet Graph and a Star Graph
The exact crossing number is only known for a small number of families of
graphs. Many of the families for which crossing numbers have been determined
correspond to cartesian products of two graphs. Here, the cartesian product of
the Sunlet graph, denoted , and the Star graph, denoted
, is considered for the first time. It is proved that the crossing
number of is , and the crossing number of
is . An upper bound for the crossing number of
is also given
On 13-Crossing-Critical Graphs with Arbitrarily Large Degrees
A recent result of Bokal et al. [Combinatorica, 2022] proved that the exact
minimum value of c such that c-crossing-critical graphs do not have bounded
maximum degree is c=13. The key to that result is an inductive construction of
a family of 13-crossing-critical graphs with many vertices of arbitrarily high
degrees. While the inductive part of the construction is rather easy, it all
relies on the fact that a certain 17-vertex base graph has the crossing number
13, which was originally verified only by a machine-readable computer proof. We
provide a relatively short self-contained computer-free proof of the latter
fact. Furthermore, we subsequently generalize the critical construction in
order to provide a definitive answer to a remaining open question of this
research area; we prove that for every c>=13 and integers d,q, there exists a
c-crossing-critical graph with more than q vertices of each of the degrees
3,4,...,d
Advances and Novel Approaches in Discrete Optimization
Discrete optimization is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a Special Issue in the journal Mathematics entitled ‘Advances and Novel Approaches in Discrete Optimization’. It contains 17 articles covering a broad spectrum of subjects which have been selected from 43 submitted papers after a thorough refereeing process. Among other topics, it includes seven articles dealing with scheduling problems, e.g., online scheduling, batching, dual and inverse scheduling problems, or uncertain scheduling problems. Other subjects are graphs and applications, evacuation planning, the max-cut problem, capacitated lot-sizing, and packing algorithms
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group