34 research outputs found
Homomorphism complexes, reconfiguration, and homotopy for directed graphs
The neighborhood complex of a graph was introduced by Lov\'asz to provide
topological lower bounds on chromatic number. More general homomorphism
complexes of graphs were further studied by Babson and Kozlov. Such `Hom
complexes' are also related to mixings of graph colorings and other
reconfiguration problems, as well as a notion of discrete homotopy for graphs.
Here we initiate the detailed study of Hom complexes for directed graphs
(digraphs). For any pair of digraphs graphs and , we consider the
polyhedral complex that parametrizes the directed graph
homomorphisms . Hom complexes of digraphs have applications
in the study of chains in graded posets and cellular resolutions of monomial
ideals. We study examples of directed Hom complexes and relate their
topological properties to certain graph operations including products,
adjunctions, and foldings. We introduce a notion of a neighborhood complex for
a digraph and prove that its homotopy type is recovered as the Hom complex of
homomorphisms from a directed edge. We establish a number of results regarding
the topology of directed neighborhood complexes, including the dependence on
directed bipartite subgraphs, a digraph version of the Mycielski construction,
as well as vanishing theorems for higher homology. The Hom complexes of
digraphs provide a natural framework for reconfiguration of homomorphisms of
digraphs. Inspired by notions of directed graph colorings we study the
connectivity of for a tournament. Finally, we use
paths in the internal hom objects of digraphs to define various notions of
homotopy, and discuss connections to the topology of Hom complexes.Comment: 34 pages, 10 figures; V2: some changes in notation, clarified
statements and proofs, other corrections and minor revisions incorporating
comments from referee
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session
Graph Theory
This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results
List covering of regular multigraphs
A graph covering projection, also known as a locally bijective homomorphism,
is a mapping between vertices and edges of two graphs which preserves
incidencies and is a local bijection. This notion stems from topological graph
theory, but has also found applications in combinatorics and theoretical
computer science.
It has been known that for every fixed simple regular graph of valency
greater than 2, deciding if an input graph covers is NP-complete. In recent
years, topological graph theory has developed into heavily relying on multiple
edges, loops, and semi-edges, but only partial results on the complexity of
covering multigraphs with semi-edges are known so far. In this paper we
consider the list version of the problem, called \textsc{List--Cover}, where
the vertices and edges of the input graph come with lists of admissible
targets. Our main result reads that the \textsc{List--Cover} problem is
NP-complete for every regular multigraph of valency greater than 2 which
contains at least one semi-simple vertex (i.e., a vertex which is incident with
no loops, with no multiple edges and with at most one semi-edge). Using this
result we almost show the NP-co/polytime dichotomy for the computational
complexity of \textsc{ List--Cover} of cubic multigraphs, leaving just five
open cases.Comment: Accepted to IWOCA 202
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum