简介:InapaperbyZhangandChenetal.(see[11]),aconjecturewasmadeconcerningtheminimumnumberofcolorsXat(G)requiredinapropertotal-coloringofGsothatanytwoadjacentverticeshavedifferentcolorsets,wherethecolorsetofavertexvisthesetcomposedofthecolorofvandthecolorsincidenttov.WefindtheexactvaluesofXat(G)andthusverifytheconjecturewhenGisaGeneralizedHalingraphwithmaximumdegreeatleast6.AgeneralizedHalingraphisa2-connectedplanegraphGsuchthatremovingalltheedgesoftheboundaryoftheexteriorfaceofG(thedegreesoftheverticesintheboundaryofexteriorfaceofGareallthree)givesatree.