1、DiscreteMathematics ( ) ContentslistsavailableatScienceDirect DiscreteMathematics journalhomepage: Note ProofofaconjectureonconnectivityofKroneckerproductofgraphs YunWang,BaoyindurengWu CollegeofMathematicsandSystemScience,XinjiangUniversity,Urumqi,Xinjiang830046,PRChina a r t i c l e i n f o Articl
2、ehistory: Received10August2010 Receivedinrevisedform1June2011 Accepted2June2011 Availableonlinexxxx Keywords: Kroneckerproduct Cartesianproduct Connectivity a b s t r a c t ForagraphG,(G)denotesitsconnectivity.TheKroneckerproductG 1 G 2 ofgraphsG 1 andG 2 isthegraphwiththevertexsetV( G 1 ) V( G 2 )
3、,twovertices( u 1 ,v 1 )and( u 2 ,v 2 ) beingadjacentinG 1 G 2 ifandonlyifu 1 u 2 E( G 1 )andv 1 v 2 E( G 2 ) .GujiandVumar R.Guji,E.Vumar,AnoteontheconnectivityofKroneckerproductsofgraphs,Appl.Math. Lett.22(2009)13601363conjecturedthatforanynontrivialgraph G,(G K n )= minn(G),( n 1)( G) whenn 3.Int
4、hisnote,weconfirmthisconjecturetobetrue. 2011ElsevierB.V.Allrightsreserved. 1. Introduction Allgraphsconsideredinthispaperarefiniteandsimple.Fornotationandterminologynotdefinedhere,wereferto West6.Letusdenotetheconnectivityandtheedge-connectivityofagraphGby(G) and ( G) ,respectively. TheCartesianpro
5、ductG 1G 2 ofgraphsG 1 andG 2 isthegraphwithV( G 1G 2 )= V( G 1 ) V( G 2 ) ,inwhichtwovertices ( u 1 ,v 1 )and( u 2 ,v 2 )areadjacentifandonlyifu 1 = u 2 andv 1 v 2 E( G 2 )orifv 1 = v 2 andu 1 u 2 E( G 1 ) .pacapan4 provedthatforanynontrivialgraphsGandH, (G 1G 2 )=min (G 1 ) |G 2 |,(G 2 ) |G 1 |,(G
6、 1G 2 ) . XuandYang7provedthat ( G 1G 2 )=min ( G 1 ) |G 2 |, ( G 2 ) |G 1 |,(G 1G 2 ) . TheKroneckerproductG 1 G 2 ofgraphsG 1 andG 2 isthegraphwiththevertexsetV( G 1 ) V( G 2 ) ,inwhichtwovertices ( u 1 ,v 1 ) and( u 2 ,v 2 ) areadjacentifandonlyifu 1 u 2 E( G 1 ) andv 1 v 2 E( G 2 ) .Hence,itiscl
7、earthatthedegreeofavertex ( u,v) inG 1 G 2 isequaltod G 1 ( u) d G 2 (v) . Weichesel5provedthatifG 1 andG 2 aretwoconnectedgraphs,thenG 1 G 2 isconnectedifandonlyifG 1 andG 2 arenotbothbipartitegraphs.AlthoughtherearemanypapersontheKroneckerproduct(sometimescalleddirectproduct, tensorproduct,crosspr
8、oduct,categoricalproduct,orconjunction,etc.)ofgraphs,veryfewresultsontheconnectivityofthe Kroneckerproductofgraphshavebeenreported.Brearandpacapan1obtainedsomeboundsontheedge-connectivity ofKroneckerproductsofgraphs,andupperboundsontheconnectivityoftheKroneckerproductsofgraphs.Mamutand Vumar3showedt
9、hat(K n K m )= ( n 1) ( m 1) foranyn m 2andn 3.Veryrecently,GujiandVumar2 showedthatforanyconnectedbipartitegraphGandcompletegraphK n withn 3,(G K n )=minn(G),( n 1)( G) . Moreover,theyputforwardthefollowingconjecture. ResearchsupportedbytheKeyProjectofChineseMinistryofEducation(No.210243). Correspondingauthor. E-mailaddresses:,(B.Wu). 0012-365X/$seefrontmatter 2011ElsevierB.V.Allrightsreserved. doi:10.1016/j.disc.2011.06.001
