{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T08:47:16Z","timestamp":1778575636699,"version":"3.51.4"},"reference-count":19,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T00:00:00Z","timestamp":1665964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CSIR Junior Research Fellowship","award":["09\/942 (0020)\/2020-EMR-I"],"award-info":[{"award-number":["09\/942 (0020)\/2020-EMR-I"]}]},{"name":"Research Group in Mathematics and Applied Mathematics, Chiang Mai University","award":["09\/942 (0020)\/2020-EMR-I"],"award-info":[{"award-number":["09\/942 (0020)\/2020-EMR-I"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Total Coloring of a graph G is a type of graph coloring in which any two adjacent vertices, an edge, and its incident vertices or any two adjacent edges do not receive the same color. The minimum number of colors required for the total coloring of a graph is called the total chromatic number of the graph, denoted by \u03c7\u2033(G). Mehdi Behzad and Vadim Vizing simultaneously worked on the total colorings and proposed the Total Coloring Conjecture (TCC). The conjecture states that the maximum number of colors required in a total coloring is \u0394(G)+2, where \u0394(G) is the maximum degree of the graph G. Graphs derived from the symmetric groups are robust graph structures used in interconnection networks and distributed computing. The TCC is still open for the circulant graphs. In this paper, we derive the upper bounds for \u03c7\u2033(G) of some classes of Cayley graphs on non-abelian groups, typically Cayley graphs on the symmetric groups and dihedral groups. We also obtain the upper bounds of the total chromatic number of complements of Kneser graphs.<\/jats:p>","DOI":"10.3390\/sym14102173","type":"journal-article","created":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T05:08:02Z","timestamp":1665983282000},"page":"2173","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Total Coloring of Some Classes of Cayley Graphs on Non-Abelian Groups"],"prefix":"10.3390","volume":"14","author":[{"given":"Shantharam","family":"Prajnanaswaroopa","sequence":"first","affiliation":[{"name":"Department of Mathematics, Amrita School of Physical Sciences, Amrita Vishwavidyapeetham, Coimbatore 641112, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jayabalan","family":"Geetha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Amrita School of Physical Sciences, Amrita Vishwavidyapeetham, Coimbatore 641112, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2226-1845","authenticated-orcid":false,"given":"Kanagasabapathi","family":"Somasundaram","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Amrita School of Physical Sciences, Amrita Vishwavidyapeetham, Coimbatore 641112, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1239-5586","authenticated-orcid":false,"given":"Teerapong","family":"Suksumran","sequence":"additional","affiliation":[{"name":"Research Group in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,17]]},"reference":[{"key":"ref_1","unstructured":"Behzad, M. (1965). Graphs and Their Chromatic Numbers. [Ph.D. Thesis, Michigan State University]."},{"key":"ref_2","first-page":"117","article-title":"Some unsolved problems in graph theory. UspekhiMat","volume":"23","author":"Vizing","year":"1968","journal-title":"Nauk"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/0012-365X(92)00058-Y","article-title":"Total coloring regular bipartite graphs is NP-hard","volume":"124","author":"McDiarmid","year":"1994","journal-title":"Discret. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/0012-365X(89)90187-8","article-title":"Determining the total colouring number is NP-hard","volume":"78","year":"1989","journal-title":"Discret. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Yap, H.P. (1996). Total Colourings of Graphs, Springer. Lecture Notes in Mathematics, 1623.","DOI":"10.1007\/BFb0092895"},{"key":"ref_6","first-page":"180","article-title":"On the total colouring of planar graphs","volume":"394","author":"Borodin","year":"1989","journal-title":"J. Reine Angew. Math."},{"key":"ref_7","unstructured":"Geetha, J., Narayanan, N., and Somasundaram, K. (2022, September 06). Total Colorings-A Survey. Available online: https:\/\/arxiv.org\/abs\/1812.05833."},{"key":"ref_8","unstructured":"Basavaraju, M., Chandran, L.S., Francis, M.C., and Naskar, A. (2021). Weakening Total Coloring Conjecture: Weak TCC and Hadwiger\u2019s Conjecture on Total Graphs. arXiv."},{"key":"ref_9","unstructured":"Conrad, K. Generating Sets. Personal communication."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1007\/s11227-011-0614-4","article-title":"Mutually independent Hamiltonian cycles in alternating group graphs","volume":"61","author":"Su","year":"2012","journal-title":"J. Supercomput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.dam.2006.08.010","article-title":"A result on the total colouring of powers of cycles","volume":"155","author":"Campos","year":"2007","journal-title":"Discret. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1016\/S0893-9659(02)00042-3","article-title":"Behzad-Vizing conjecture and Cartesian-product graphs","volume":"15","author":"Zmazek","year":"2002","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2150050","DOI":"10.1142\/S1793830921500506","article-title":"Total colorings of circulant graphs","volume":"13","author":"Geetha","year":"2021","journal-title":"Discret. Math. Algorithms Appl."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Navaneeth, R., Geetha, J., Somasundaram, K., and Fu, H.-L. (2022). Total colorings of some classes of four regular circulant graphs. AKCE Int. J. Graphs Comb., 1\u20133.","DOI":"10.1080\/09728600.2022.2088316"},{"key":"ref_15","first-page":"667","article-title":"On total coloring of some classes of regular graphs","volume":"1","author":"Prajnanaswaroopa","year":"2022","journal-title":"Taiwan J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zorzi, A., Figuiredo, C.M.H., Machado, R.C.S., Zatesko, L.M., and Souza, U.S. (2021). Compositions, decompositions, and conformability for total coloring on power of cycle graphs. Discret. Appl. Math., in press.","DOI":"10.1016\/j.dam.2021.06.012"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/0095-8956(85)90056-5","article-title":"On 1-factorizability of Cayley graphs","volume":"39","author":"Stong","year":"1985","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/0095-8956(79)90002-9","article-title":"The edge-coloring of complete hypergraphs I","volume":"26","author":"Baranyai","year":"1979","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_19","unstructured":"Monteil, T. (2022, September 06). Can You Find the Total Chromatic Number (Edge and Vertices) of a Graph?. Available online: https:\/\/ask.sagemath.org\/question\/35744\/can-you-find-the-total-chromatic-number-edge-and-vertices-of-a-graph\/?answer=35745."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2173\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:55:31Z","timestamp":1760144131000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2173"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,17]]},"references-count":19,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102173"],"URL":"https:\/\/doi.org\/10.3390\/sym14102173","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,17]]}}}