{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:53:37Z","timestamp":1760237617409,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T00:00:00Z","timestamp":1591315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we construct the colored-path homology theory in the category of vertex colored (di)graphs and describe its basic properties. Our construction is based on the path homology theory of digraphs that was introduced in the papers of Grigoryan, Muranov, and Shing-Tung Yau and stems from the notion of the path complex. Any graph naturally gives rise to a path complex in which for a given set of vertices, paths go along the edges of the graph. We define path complexes of vertex colored (di)graphs using the natural restrictions that are given by coloring. Thus, we obtain a new collection of colored-path homology theories. We introduce the notion of colored homotopy and prove functoriality as well as homotopy invariance of homology groups. For any colored digraph, we construct the spectral sequence of colored-path homology groups which gives the effective method of computations in the general case since any (di)graph can be equipped with various colorings. We provide a lot of examples to illustrate our results as well as methods of computations. We introduce the notion of homotopy and prove functoriality and homotopy invariance of introduced vertexed colored-path homology groups. For any colored digraph, we construct the spectral sequence of path homology groups which gives the effective method of computations in the constructed theory. We provide a lot of examples to illustrate obtained results as well as methods of computations.<\/jats:p>","DOI":"10.3390\/sym12060965","type":"journal-article","created":{"date-parts":[[2020,6,9]],"date-time":"2020-06-09T06:34:16Z","timestamp":1591684456000},"page":"965","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Path Homology of Vertex Colored (Di)Graphs"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2398-8093","authenticated-orcid":false,"given":"Yuri V.","family":"Muranov","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, S\u0142oneczna 54, 10-710 Olsztyn, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anna","family":"Szczepkowska","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, S\u0142oneczna 54, 10-710 Olsztyn, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"179","DOI":"10.4310\/HHA.2018.v20.n2.a9","article-title":"On the path homology theory of digraphs and Eilenberg-Steenrod axioms","volume":"20","author":"Jimenez","year":"2018","journal-title":"Homol. Homotopy Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"619","DOI":"10.4310\/PAMQ.2014.v10.n4.a2","article-title":"Homotopy theory for digraphs","volume":"10","author":"Lin","year":"2014","journal-title":"Pure Appl. Math. Q."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"887","DOI":"10.4310\/AJM.2015.v19.n5.a5","article-title":"Cohomology of digraphs and (undirected) graphs","volume":"19","author":"Lin","year":"2015","journal-title":"Asian J. Math."},{"key":"ref_4","first-page":"1319","article-title":"Path homology theory of multigraphs and quivers","volume":"5","author":"Muranov","year":"2018","journal-title":"Forum Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"295","DOI":"10.4310\/HHA.2014.v16.n1.a16","article-title":"Graphs associated with simplicial complexes","volume":"16","author":"Muranov","year":"2014","journal-title":"Homol. Homotopy Appl."},{"key":"ref_6","first-page":"209","article-title":"On a cohomology of digraphs and Hochschild cohomology","volume":"11","author":"Muranov","year":"2015","journal-title":"J. Homotopy Relat. Struct."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Lowell, W., Bieneke, E., and Wilson, R.J. (2015). Topics in Chromatic Graph Theory, Cambridge University Press. Enyclopedia of Mathematics and Its Applications 156.","DOI":"10.1017\/CBO9781139519793"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.dam.2010.11.004","article-title":"Improper C-colorings of graphs","volume":"159","author":"Sampathkumar","year":"2011","journal-title":"Discret. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Chartrand, G., Lesniak, L., and Zhang, P. (2011). Graphs and Digraphs, CRC Press.","DOI":"10.1201\/b14892"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hell, P., and Ne\u0161et\u0159il, J. (2004). Graphs and Homomorphisms, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198528173.001.0001"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1002\/jgt.3190100207","article-title":"Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency","volume":"10","author":"Cowen","year":"1986","journal-title":"Graph Theory"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Lang, S. (2002). Algebra, Springer. [3rd ed.]. Graduate Texts in Mathematics.","DOI":"10.1007\/978-1-4613-0041-0"},{"key":"ref_13","unstructured":"MacLane, S. (1963). Homology, Springer. Die Grundlehren der Mathematischen Wissenschaften. Bd. 114."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3197","DOI":"10.1088\/0305-4470\/28\/11\/019","article-title":"Differential calculi on commutative algebras","volume":"28","author":"Baehr","year":"1995","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1112\/blms\/bdu043","article-title":"Discrete homology theory for metric spaces","volume":"46","author":"Barcelo","year":"2014","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3159","DOI":"10.1088\/0305-4470\/27\/9\/028","article-title":"Differential calculus and gauge theory on finite sets","volume":"27","author":"Dimakis","year":"1994","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6703","DOI":"10.1063\/1.530638","article-title":"Discrete differential calculus: Graphs, topologies, and gauge theory","volume":"35","author":"Dimakis","year":"1994","journal-title":"J. Math. Phys."},{"key":"ref_18","unstructured":"Mosher, R.E., and Tangora, M.C. (1968). Cohomology Operations and Applications in Homotopy Theory, Harper & Row, Publishers."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Spanier, E.H. (1966). Algebraic Topology, MeGraw-Hill.","DOI":"10.1007\/978-1-4684-9322-1_5"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1006\/aama.2000.0710","article-title":"Foundations of a connectivity theory for simplicial complexes","volume":"26","author":"Barcelo","year":"2001","journal-title":"Adv. Appl. Math."},{"key":"ref_21","unstructured":"Connes, A. (1994). Noncommutative Geometry, Academic Press."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10801-006-9100-0","article-title":"Homotopy theory of graphs","volume":"24","author":"Babson","year":"2006","journal-title":"J. Algebr. Comb."},{"key":"ref_23","first-page":"35","article-title":"Fundamental groupoid of digraphs and graphs","volume":"143","author":"Jimenez","year":"2018","journal-title":"Czechoslov. Math. J."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/965\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:35:58Z","timestamp":1760175358000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/965"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,5]]},"references-count":23,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["sym12060965"],"URL":"https:\/\/doi.org\/10.3390\/sym12060965","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,6,5]]}}}