{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T05:40:16Z","timestamp":1698385216657},"reference-count":13,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,10,6]],"date-time":"2006-10-06T00:00:00Z","timestamp":1160092800000},"content-version":"vor","delay-in-days":4023,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1995,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A graph <jats:italic>G<\/jats:italic> is degree\u2010bounded\u2010colorable (briefly, db\u2010colorable) if it can be properly vertex\u2010colored with colors 1,2, \u2026, k \u2264 \u0394(<jats:italic>G<\/jats:italic>) such that each vertex <jats:italic>v<\/jats:italic> is assigned a color <jats:italic>c<\/jats:italic>(<jats:italic>v<\/jats:italic>) \u2264 <jats:italic>v<\/jats:italic>. We first prove that if a connected graph <jats:italic>G<\/jats:italic> has a block which is neither a complete graph nor an odd cycle, then <jats:italic>G<\/jats:italic> is db\u2010colorable. One may think of this as an improvement of Brooks' theorem in which the global bound \u0394(<jats:italic>G<\/jats:italic>) on the number of colors is replaced by the local bound deg <jats:italic>v<\/jats:italic> on the color at vertex <jats:italic>v<\/jats:italic>. Extending the above result, we provide an algorithmic characterization of db\u2010colorable graphs, as well as a nonalgorithmic characterization of db\u2010colorable trees. We briefly examine the problem of determining the smallest integer <jats:italic>k<\/jats:italic> such that <jats:italic>G<\/jats:italic> is db\u2010colorable with colors 1, 2,\u2026, <jats:italic>k<\/jats:italic>. Finally, we extend these results to set coloring. \u00a9 1995, John Wiley &amp; Sons, Inc.<\/jats:p>","DOI":"10.1002\/jgt.3190200207","type":"journal-article","created":{"date-parts":[[2007,6,7]],"date-time":"2007-06-07T16:22:39Z","timestamp":1181233359000},"page":"177-194","source":"Crossref","is-referenced-by-count":2,"title":["Degree\u2010bounded coloring of graphs: Variations on a theme by brooks"],"prefix":"10.1002","volume":"20","author":[{"given":"S. L.","family":"Hakimi","sequence":"first","affiliation":[]},{"given":"J.","family":"Mitchem","sequence":"additional","affiliation":[]},{"given":"E. F.","family":"Schmeichel","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,6]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"69","article-title":"Cliques, colorings, and locallyperfect graphs","volume":"39","author":"Albertson M. O.","year":"1983","journal-title":"Congress. 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