{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:46:34Z","timestamp":1759063594678,"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>A $t$-tone $k$-coloring of $G$ assigns to each vertex of $G$ a set of $t$ colors from $\\{1,\\dots,k\\}$\u00a0so that vertices at distance $d$ share fewer than $d$ common colors. The $t$-tone chromatic\u00a0number of $G$, denoted $t(G)$, is the minimum $k$ such that $G$ has a $t$-tone $k$-coloring.\u00a0Bickle and Phillips showed that always $\\tau_2(G) \\leq [\\Delta(G)]^2\u00a0+\\Delta(G)$, but conjectured that\u00a0in fact $\\tau_2(G) \\leq 2\\Delta(G) + 2$; we confirm this conjecture when $\\Delta(G) \\leq 3$ and also show\u00a0that always $\\tau_2(G) \\leq \\lceil\u00a0(2 +\\sqrt{2}) \\Delta(G) \\rceil$. For general $t$ we prove that $\\tau_t(G) \\leq (t^2+t) \\Delta(G)$.\u00a0Finally, for each $t \\geq 2$ we show that there exist constants $c_1$\u00a0and $c_2$\u00a0such that for every\u00a0tree $T$ we have $c_1 \\sqrt{\\Delta(T)} \\leq \\tau_t(T) \\leq c_2\\sqrt{\\Delta(T)}$.<\/jats:p>","DOI":"10.37236\/2710","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:13:11Z","timestamp":1578719591000},"source":"Crossref","is-referenced-by-count":4,"title":["New Results in t-Tone Coloring of Graphs"],"prefix":"10.37236","volume":"20","author":[{"given":"Daniel","family":"Cranston","sequence":"first","affiliation":[]},{"given":"Jaehoon","family":"Kim","sequence":"additional","affiliation":[]},{"given":"William","family":"Kinnersley","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,4,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p17\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:21:11Z","timestamp":1579260071000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i2p17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,24]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2013,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/2710","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,4,24]]},"article-number":"P17"}}