{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:28Z","timestamp":1753893808340,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The Total Colouring Conjecture suggests that $\\Delta+3$ colours ought to suffice in order to provide a proper total colouring of every graph $G$ with maximum degree $\\Delta$. Thus far this has been confirmed up to an additive constant factor, and the same holds even if one additionally requires every pair of neighbours in $G$ to differ with respect to the sets of their incident colours, so called pallets. Within this paper we conjecture that an upper bound of the form $\\Delta+C$, for a constant $C&gt;0$ still remains valid even after extending the distinction requirement to pallets associated with vertices at distance at most $r$, if only $G$ has minimum degree $\\delta$ larger than a constant dependent on $r$. We prove that such assumption on $\\delta$ is then unavoidable and exploit the probabilistic method in order to provide two supporting results for the conjecture. Namely, we prove the upper bound $(1+o(1))\\Delta$ for every $r$, and show that\u00a0for any fixed $\\epsilon\\in(0,1]$ and $r$, the conjecture holds if $\\delta\\geq \\varepsilon\\Delta$, i.e., in particular for regular graphs.<\/jats:p>","DOI":"10.37236\/5481","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:53:54Z","timestamp":1578689634000},"source":"Crossref","is-referenced-by-count":1,"title":["Distant Set Distinguishing Total Colourings of Graphs"],"prefix":"10.37236","volume":"23","author":[{"given":"Jakub","family":"Przyby\u0142o","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,6,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i2p54\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i2p54\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:27:26Z","timestamp":1579238846000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i2p54"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6,24]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2016,3,31]]}},"URL":"https:\/\/doi.org\/10.37236\/5481","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,6,24]]},"article-number":"P2.54"}}