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TheUniversityofSydney
SchoolofMathematicsandStatistics
Tutorial2(Week9)
Moredi cultquestionsaremarkedwitheither*or**.Thosemarked*areatthelevelwhichMATH2069studentswillhavetosolveinordertobesureofgettingaCredit,ortohaveachanceofaDistinctionorHighDistinction.Thosemarked**aremainlyintendedforMATH2969students.
1.Determinewhethereachofthefollowingsequencesisthedegreesequenceofa
graph.Ifso,drawapictureofsuchagraph.
(a)(2,2,2)(b)(0,1,2,3)(c)(2,3,3,4,4,5)(d)(2,3,4,4,5)
(e)(1,1,1,1,4)(f)(1,3,3,3)(g)(1,2,2,3,4,4)(h)(1,3,3,4,5,6,6)
2.SupposethatGhas23vertices,andeveryvertexhasdegreeatleast11.
(a)IsitpossiblethatGisregularofdegree11?
(b)ProvethatGisconnected.
3.Foreachofthefollowinggraphs,either ndanEuleriancircuit, ndanEulerian
trail,orexplainwhyneitherexists.
(a)
(b)
e
b
*(d)
(c)
4.Findthenumberofvertices,thenumberofedges,andthedegreesof:
(a)thecompletebipartitegraphKp,q;
(b)thecubegraphQm.
*5.Whatisthesmallestnumbernforwhichthereexisttwonon-isomorphicgraphs
withnverticeshavingthesamedegreesequence?
c2014TheUniversityofSydneyCopyright
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