Several Labelling Problems Of Graphs

A k-(d,1)-total labelling of a graph G is a mapping from V(G)?E(G)to the label set {0,1,...,k} such that any two adjacent vertices have different labels,any two adjacent edges have different labels and any incident vertex and edge have the label difference at least d.The(d,1)-total labelling number ?dT(G)of a graph G is the least k such that G has a k-(d,1)-total labelling.A k-(2,1)-coupled labelling of a plane graph G is a mapping from B(G)?G(G)to the label set(0,1,...,k} such that any two adjacent vertices have different labels,any two adjacent faces have different labels and any incident vertex and face have the label difference at least 2.The(2,1)-coupled labelling number ?2vf(G)of a plane graph G is the least k such that G has a k-(2,1)-coupled labelling.In 2002,Havet and Yu[7]investigated the(d,1)-total labelling of graphs and conjectured that:For any graph G,?dT(G)??(G)+2d-1.In 2015,Chen[20]investigated the(2,1)-coupled labelling of plane graphs and conjectured that:For any plane graph G,?2vf(G)?7.In this thesis,we study(2,1)-total labelling and(2,1)-coupled labelling problems of some graphs.The thesis consists of three parts as follows.In Chapter 1,we collect some basic notions used in the thesis and give a chief survey on two kinds of labelling problems for graphs.Main results in the thesis are stated.In Chapter 2,we discuss the(2,1)-coupled labelling of some special graphs such as outerplane graphs,Halin graphs,Prism graphs,etc.In Chapter 3,we discuss the(2,1)-total labelling of some special graphs such as Halin graphs,general Mycielski graph,M(Kn,m)graphs,etc.