{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:03Z","timestamp":1753893783258,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Dirac's classic theorem asserts that if ${\\bf G}$ is a graph on $n$ vertices, and $\\delta({\\bf G})\\ge n\/2$, then ${\\bf G}$ has a hamilton cycle.  As is well known, the proof also shows that if $\\deg(x)+\\deg(y)\\ge(n-1)$, for every pair $x$, $y$ of independent vertices in ${\\bf G}$, then ${\\bf G}$ has a hamilton path.  More generally, S. Win has shown that if $k\\ge 2$, ${\\bf G}$ is connected and $\\sum_{x\\in I}\\deg(x)\\ge n-1$ whenever $I$ is a $k$-element independent set, then ${\\bf G}$ has a spanning tree ${\\bf T}$ with $\\Delta({\\bf T})\\le k$. Here we are interested in the structure of spanning trees under the additional assumption that ${\\bf G}$ does not have a spanning tree with maximum degree less than $k$. We show that apart from a single exceptional class of graphs, if $\\sum_{x\\in I}\\deg(x)\\ge n-1$ for every $k$-element independent set, then ${\\bf G}$ has a spanning caterpillar ${\\bf T}$ with maximum degree $k$.  Furthermore, given a maximum path $P$ in ${\\bf G}$, we may require that $P$ is the spine of ${\\bf T}$ and that the set of all vertices whose degree in ${\\bf T}$ is $3$ or larger is independent in ${\\bf T}$.<\/jats:p>","DOI":"10.37236\/1577","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:07:02Z","timestamp":1578708422000},"source":"Crossref","is-referenced-by-count":8,"title":["Spanning Trees of Bounded Degree"],"prefix":"10.37236","volume":"8","author":[{"given":"Andrzej","family":"Czygrinow","sequence":"first","affiliation":[]},{"given":"Genghua","family":"Fan","sequence":"additional","affiliation":[]},{"given":"Glenn","family":"Hurlbert","sequence":"additional","affiliation":[]},{"given":"H. A.","family":"Kierstead","sequence":"additional","affiliation":[]},{"given":"William T.","family":"Trotter","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2001,10,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r33\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r33\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:16:53Z","timestamp":1579324613000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v8i1r33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,10,2]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2001,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1577","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2001,10,2]]},"article-number":"R33"}}