We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call ‘spider moves.’ We then create a category by modding out by the 2-cells of our 2-category and use the spider moves to show that for finite graphs, this category is a homotopy category in the sense that it satisfies the universal property for localizing homotopy equivalences. We then show that finite stiff graphs form a skeleton of this homotopy category.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Babson, E., Barcelo, H., de Loungeville, M., Laubenbacher, R.: Homotopy theory of graphs. J. Algebraic Combin. 24(1), 31–44 (2006)
Babson, E., Kozlov, D.N.: Proof of the Lovàsz conjecture. Ann. Math. 165(3), 965–1007 (2007)
Bonato, A., Nowakowski, R.: The Game of Cops and Robbers on Graphs. Student mathematical library. American Mathematical Society, New York (2010)
Bondy, J., Murty, U.: Graph theory. 2008. Grad. Texts Math (2008)
Brightwell, G.R., Winkler, P.: Gibbs measures and dismantlable graphs. J. Comb. Theory Ser. B 78(1), 141–66 (2000)
Dochtermann, A.: Hom complexes and homotopy theory in the category of graphs. European J. Combin. 30(2), 490–509 (2009)
Dochtermann, A.: Homotopy groups of hom complexes of graphs. J. Combin. Theory Ser. A 116(1), 180–194 (2009)
Droz, J.M.: Quillen model structures on the category of graphs. Homology Homotopy Appl. 14(2), 265–284 (2012)
Fieux, E., Lacaze, J.: Foldings in graphs and relations with simplicial complexes and posets. Discrete Math. 312(17), 2639–2651 (2012)
Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics, 1st edn. Springer, New York (2001)
Goyal, S., Santhanam, R.: (Lack of) Model structures on the category of graphs (2019). arXiv:1902.09182
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001)
Hell, P., Nešetřil, J.: The core of a graph. Discrete Math. 109, 117–126 (1992)
Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford Lecture Series in Mathematics and its Applications, vol. 28. Oxford University Press, Oxford (2004)
Kozlov, D.N.: Chromatic numbers, morphism complexes, and Stiefel-Whitney characteristic classes. In: Geometric Combinatorics, vol. 13, pp. 249–315. American Mathematical society, Providence, RI (2007)
Kozlov, D.N.: Collapsing along monotone poset maps. Int. J. Math. 8, 56 (2006)
Kozlov, D.N.: Simple homotopy types of Hom-complexes, neighborhood complexes, Lovàsz complexes, and atom crosscut complexes. Topology Appl. 153(14), 2445–2454 (2006)
Kozlov, D.N.: A simple proof for folds on both sides in complexes of graph homomorphisms. Proc. Amer. Math. Soc. 134(5), 1265–1270 (2006)
Mac, L.S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, New York (1998)
Matsushita, T.: Box complexes and homotopy theory of graphs. Homology Homotopy Appl. 19(2), 175–197 (2017)
Plessas, D.: The Categories of Graphs. Ph.D. thesis, The University of Montana (2012)
Riehl, E.: Category Theory in Context. Dover Modern Math Originals. Dover Publications, Aurora (2017)
The authors are grateful to Dr. Demitri Plessas for his previous forays into categorical graph theory and for his feedback. We also want to thank Dr. Jeffery Johnson for helping us with some of the terminology in this paper. Lastly, we want to thank the referees who pointed us in the direction of the relevant literature.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Chih, T., Scull, L. A homotopy category for graphs. J Algebr Comb (2020). https://doi.org/10.1007/s10801-020-00960-5
- Graph homomorphism
- Homotopy category
- Skeleton of a category
Mathematics Subject Classification