(Lack of) Model Structures on the Category of Graphs

Goyal, Shuchita; Santhanam, Rekha

doi:10.1007/s10485-021-09630-4

Cited by 5 publications

(4 citation statements)

References 8 publications

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“…In [4], different model category structures have been constructed on the category of graphs. In [5] we have shown that there is no model structure on G with ×-homotopy equivalences as the class of weak equivalences, irrespective of the choice for the class of cofibrations and fibrations. The proof makes use of the fact that in a model category, acyclic cofibrations have left lifting property with respect to fibrations.…”

Section: Introductionmentioning

confidence: 94%

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Study of Cofibration Category structures on finite Graphs

Goyal¹,

Santhanam²

2023

Preprint

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Section: Introductionmentioning

confidence: 94%

“…In the previous section we showed that (G, ×) is not a cofibration category with quasi-cofibrations. These were defined based on the analysis in [5] where we worked with a subclass of induced inclusions. We now show that removing these restrictions does not help us create a cofibration category structure on (G, ×).…”

Section: Enlarging the Class Of Cofibrationsmentioning

confidence: 99%

Study of Cofibration Category structures on finite Graphs

Goyal¹,

Santhanam²

2023

Preprint

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“…In [33] Droz constructed model categories with other notions of weak equivalences. On the other hand in [39] Goyal and Santhanam showed no model category structure exists if one takes ×-homotopy as weak equivalence and inclusions as cofibrations.…”

Section: Tournaments and Reconfigurationmentioning

confidence: 99%

Homomorphism complexes, reconfiguration, and homotopy for directed graphs

Dochtermann¹,

Singh²

2021

Preprint

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The neighborhood complex of a graph was introduced by Lovász 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. For any pair of directed graphs 𝐺 and 𝐻 we consider the polyhedral complex − −− → Hom(𝐺, 𝐻) that parametrizes the directed graph homomorphisms 𝑓 : 𝐺 → 𝐻. This construction can be seen as a special case of the poset structure on the set of multihomomorphisms in more general categories, as introduced by Kozlov, Matsushita, and others. Hom complexes of directed graphs 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 directed graph 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 directed graph version of the Mycielksi construction, as well as vanishing theorems for higher homology. The − −− → Hom complexes of directed graphs provide a natural framework for reconfiguration of homomorphisms of directed graphs. Inspired by notions of directed graph colorings we study the connectivity of − −− → Hom(𝐺, 𝑇 𝑛 ) for 𝑇 𝑛 a tournament, obtaining a complete answer for the case of transitive 𝑇 𝑛 . If 𝐺 = 𝑇 𝑚 is also a transitive tournament we describe a connection to mixed subdivisions of dilated simplices. Finally we use paths in the internal hom objects of directed graphs to define various notions of homotopy, and discuss connections to the topology of − −− → Hom complexes.

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“…In [9], a fibration category structure is constructed on the category of simple graphs in which the weak equivalences are the weak homotopy equivalences of discrete homotopy theory [3]. On the other hand, in [13], it is proven that no model category structure exists in the category of undirected graphs with loops in which weak equivalences are the ×-homotopy equivalences of Dochtermann [12] and cofibrations are a subclass of monomorphisms.…”

Section: Introductionmentioning

confidence: 99%