Semiregular factorization of simple graphs

• Main Authors: Anthony J. Hilton, Jerzy Wojciechowski
• Format: Article
• Language: English
• Published: Università degli Studi di Catania 2004-11-01
• Series: Le Matematiche
• Online Access: http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/168
• ISSN: 0373-3505; 2037-5298

<p>A graph <em>G</em> is a <em>(d, d + s)</em>-graph if the degree of each vertex of <em>G</em> lies in the interval<em> [d, d + s]</em>. A <em>(d, d + 1)</em>-graph is said to be semiregular. An <em>(r, r + 1)</em> -factorization of a graph is a decomposition of the graph into edgedisjoint <em>(r, r</em> <em>+ 1)-</em>factors.<br />We discuss here the state of knowledge about <em>(r, r + 1)</em>-factorizations of <em>d</em> -regular graphs and of <em>(d, d + 1)</em>-graphs.<br />For <em>r, s ≥ 0</em>, let <em>φ(r, s)</em> be the least integer such that, if<em> d ≥ φ(r, s)</em> and <em>G</em> is any simple <em>[d, d + s]</em>-graph, then <em>G</em> has an <em>(r, r + 1)</em>-factorization.<br />Akiyama and Kano (when r is even) and Cai (when r is odd) showed that <em>φ(r, s)</em> exists for all <em>r, s</em>. We show that, for<em> s ≥ 2, φ(r, s) = r(r + s + 1) + 1</em>. Earlier <em>φ(r,</em> <em>0)</em> was determined by Egawa and Era, and <em>φ(r, 1)</em> was determined by Hilton.</p>